Bunning, who suffered the stroke Tuesday night in his Southgate, Kentucky, home, was moved from intensive care to a transitional care unit on Thursday night, per the report.
Bunning "has been provided skilled care that is leading him on the road to recovery," the family said in a statement Friday.
"The Bunning family wants to thank the first responders and medical personnel who have been treating dad," the statement said. "We sincerely appreciate the thoughts and prayers of all who are concerned about our father’s health. However, so we can focus our efforts on dad’s recovery, we ask the press to respect our family’s privacy at this time. We will let everyone know as his health continues to improve."
The 84-year old is one of two Phillies pitchers to toss a perfect game in the organization’s history. He accomplished the feat on Father’s Day in 1964.
Along with the Phillies, Bunning played for the Tigers, Pirates and Dodgers in his 17-year career. The righthander, who was enshrined on the Phillies Wall of Fame in 1984, won 89 games and posted a 2.93 ERA in six seasons in Philadelphia.
After his baseball days, Bunning started a career in politics. He served stints in Congress and the U.S. Senate before retiring in 2010.
CHICAGO -- Cursed by a Billy Goat, bedeviled by Bartman and crushed by decades of disappointment, the Chicago Cubs are at long last headed back to the World Series.
Kyle Hendricks outpitched Clayton Kershaw, Anthony Rizzo and Willson Contreras homered early and the Cubs won their first pennant since 1945, beating the Los Angeles Dodgers 5-0 Saturday night in Game 6 of the NL Championship Series.
The drought ended when closer Aroldis Chapman got Yasiel Puig to ground into a double play, setting off a wild celebration inside Wrigley Field, outside the ballpark and all over the city.
Seeking their first crown since 1908, manager Joe Maddon's team opens the World Series at Cleveland on Tuesday night. The Indians haven't won it all since 1948 - Cleveland and Cubs have the two longest title waits in the majors.
"This city deserves it so much," Rizzo said. "We got four more big ones to go, but we're going to enjoy this. We're going to the World Series. I can't even believe that."
All-everything Javier Baez and pitcher Jon Lester shared the NLCS MVP. Baez hit .318, drove in five runs and made several sharp plays at second base. Lester, a former World Series champion in Boston, was 1-0 with a 1.38 ERA in two starts against the Dodgers.
Deemed World Series favorites since opening day, the Cubs topped the majors with 103 wins to win the NL Central, then beat the Giants and Dodgers in the playoffs.
The Cubs overcame a 2-1 deficit against the Dodgers and won their 17th pennant. They had not earned a World Series trip since winning a doubleheader opener 4-3 at Pittsburgh on Sept. 29, 1945, to clinch the pennant on the next-to-last day of the season.
The eternal "wait till next year" is over. No more dwelling on a history of failure - the future is now.
"We're too young. We don't care about it," star slugger Kris Bryant said. "We don't look into it. This is a new team, this is a completely different time of our lives. We're enjoying it and our work's just getting started."
Hendricks pitched two-hit ball for 7 1/3 innings. Chapman took over and closed with hitless relief, then threw both arms in the air as he was mobbed by teammates and coaches.
The crowd joined in, chanting and serenading their team.
"Chicago!" shouted popular backup catcher David Ross.
The Cubs shook off back-to-back shutout losses earlier in this series by pounding the Dodgers for 23 runs to win the final three games.
And they were in no way overwhelmed by the moment on Saturday, putting aside previous frustration.
In 1945, the Billy Goat Curse supposedly began when a tavern owner wasn't allowed to bring his goat to Wrigley. In 2003, the Cubs lost the final three games of the NLCS to Florida, punctuated with a Game 6 defeat when fan Steve Bartman deflected a foul ball.
Even as recently as 2012, the Cubs lost 101 times.
This time, no such ill luck.
Bryant had an RBI single and scored in a two-run first. Dexter Fowler added two hits, drove in a run and scored one.
Contreras led off the fourth with a homer. Rizzo continued his resurgence with a solo drive in the fifth.
That was plenty for Hendricks, the major league ERA leader.
Hendricks left to a standing ovation after Josh Reddick singled with one out in the eighth. The only other hit Hendricks allowed was a single by Andrew Toles on the game's first pitch.
Kershaw, dominant in Game 2 shutout, gave up five runs and seven hits before being lifted for a pinch hitter in the sixth. He fell to 4-7 in the postseason.
The Dodgers haven't been to the World Series since winning in 1988.
Pitching on five days' rest, the three-time NL Cy Young Award winner threw 30 pitches in the first. Fowler led off with a double, and Bryant's single had the crowd shaking the 102-year-old ballpark.
They had more to cheer when left fielder Andrew Toles dropped Rizzo's fly, putting runners on second and third, and Ben Zobrist made it 2-0 a sacrifice fly.
The Cubs added a run in the second when Addison Russell doubled to deep left and scored on a two-out single by Fowler.
Maddon benched slumping right fielder Jason Heyward in favor of Albert Almora Jr.
"Kershaw's pitching, so I wanted to get one more right-handed bat in the lineup, and also with Albert I don't feel like we're losing anything on defense," Maddon said. "I know Jason's a Gold Glover, but I think Albert, given an opportunity to play often enough would be considered a Gold Glove-caliber outfielder, too."
Heyward was 2 for 28 in the playoffs - 1 for 16 in the NLCS.
Kerry Wood, wearing a Ron Santo jersey, threw out the first pitch and actor Jim Belushi delivered the "Play Ball!" call before the game. Pearl Jam front man Eddie Vedder and actor John Cusack were also in attendance. And Bulls great Scottie Pippen led the seventh-inning stretch.