CHICAGO -- After receiving a one-day reprieve on Saturday from the constant Jake Peavy questions, White Sox manager Ozzie Guillen was asked what the right-hander was up to on Sunday morning during his pregame meeting with the media.

"He's having a nice breakfast or lunch down there," said Guillen with a smile, drawing a laugh from the assembled media.

In reality, that breakfast or lunch for Peavy preceded a bullpen session, marking his first throwing off the mound since he took a Wes Timmons line drive off his pitching elbow during a Minor League rehab start on Aug. 24. Peavy threw close to 60 pitches Sunday and said that his elbow felt a great deal better since he left his start in Norfolk on Aug. 29 after 3 1/3 innings due to residual pain and inflammation in that elbow.

But Peavy isn't quite ready to pronounce himself ready to take the mound in a big league contest.

"I'm getting better," said Peavy. "I'd be lying if I told you I feel completely normal, but nothing crazy. I'm able to get some extension.

"Big improvement from the start I tried to make, just because we've had time and countless numbers of treatments. Hopefully we're on the right track now without any setbacks. With as crazy as this has been, who knows? I certainly would love to pitch soon."

Guillen holds out that same hope for Peavy, as does everyone in the White Sox organization. Peavy said that there's another bullpen session scheduled in a couple of days, possibly Wednesday, and then they will re-evaluate where he stands.

As for some sort of simulated game needed before Peavy starts, he left that plan in the hands of pitching coach Don Cooper.

"I really have no idea," Peavy said. "I don't know what Coop and those guys have planned, but I'm as tired and frustrated with this as I'm sure you guys are. I want to get out there soon, so we'll see what happens."

Patience and being careful still were the key words in regards to Peavy used by Guillen, who brought up how Peavy would have been pitching this weekend against Boston if not for the line drive to the elbow. Sunday's session was encouraging, with Peavy throwing all of his pitches, although not turning it up to 100 percent.

But Peavy got a good read where he stands. He seems ready for the next step in this second comeback, after working his way into form from a partially torn tendon in his right ankle that had him sidelined since June 8. Guillen hinted that a simulated game probably would be needed before Peavy gets the call.

"Because we don't have any place to put him," Guillen said. "It's not fair for the Minor League kids in the playoffs to say, 'OK, here he is, the guy is going to rehab' and not give those kids a chance to win the championship. That's not fair. He's going to pitch here and that's what we think about it.

"We'll try to figure out when he's going to pitch, and we'll see what we have. After that, we'll find out. If we're in the pennant race, we'll use him. If not, we'll shut him down."