The Italian has been out with a dislocated shoulder, but could return this term, while Martin Kelly will not be rushed back.
Italian international Borini sustained the injury against Swansea last month, but Driscoll is optimistic that the 21-year-old's recovery from surgery is ahead of schedule.
"Fabio Borini was outside for the first time today, so he's starting that progression - passing balls, running and doing ball work," Driscoll told the club's official website. "He's due to be probably another six weeks from this point on. The injury he's had is normally eight to 12 weeks. He's quite a quick rehabilitator."
Driscoll believes that Borini's fast recovery could point to a return before the end of the season.
"At the moment, it's looking pretty positive for Fabio that he could be back ahead of schedule and maybe make the end of the season," Borini stated.
Meanwhile, defender Martin Kelly will not be risked this term as he continues his rehabilitation from a serious knee injury.
Kelly ruptured the anterior cruciate ligament in his right knee during September's game against Manchester United but Driscoll says the defender should look no further than the club and national team captain for inspiration during his recovery.
"Martin is doing extremely well; he's probably at the late stage of his rehab," Driscoll said. "Martin should look at what Steven Gerrard has achieved this season; people would never have considered that he would be available for a full campaign."
He added, "He should take great hope from that and know that if he continues to work and continues his professionalism, hopefully the future is bright. We hope to give him every chance.
"We'll probably look to prioritize getting him back for preseason 110 percent fit - 10 percent fitter, stronger and quicker, more powerful and agile than he was previously."
Driscoll explained that Kelly could technically play by the end of the season but insisted that Liverpool is playing it safe and wants to see the player totally fit next season.
"He'll probably be in a position where it will be tempting to put him into games again this season. But it will show class and quality if we make sure that he's in a very strong position so he can fly next season."