The two players practised Monday, and head coach Lindy Ruff said both could be cleared for contact by Wednesday. Ruff said it's possible one or both could play before the Sabres close their regular season Sunday at Philadelphia.
"It's a big addition if we can get them ready and on top of their games," Ruff said.
Afinogenov is a dynamic and speedy first-line winger who has missed 20 games with a broken left wrist. He was among the team's leaders before he was hurt, registering 23 goals and 34 assists in 53 games.
Connolly is a play-making centre who has missed the entire season since sustaining a concussion during the second round of the playoffs last May. His headaches since cleared, Connolly experienced a setback in his recovery six weeks ago when he broke a bone below his knee.
Ruff cautioned Connolly would initially be used in a limited role - possibly as a power-play specialist - because he is still working himself into shape.
That's significant news for the Eastern Conference-leading Sabres (50-21-7), who had up to seven regulars out at the same time in February. The Sabres have only one other player hurt, forward Paul Gaustad, who isn't expected back this season after having surgery to repair a sliced tendon in his left leg.
With four games left, starting Tuesday in Pittsburgh, Buffalo is one win from clinching first place in the East. The Sabres would also clinch with one more loss by Ottawa and New Jersey.
Afinogenov said he still felt some soreness in his wrist as anticipated, but was happy to finally be back practising.
"Definitely, it's a big step for me," Afinogenov said. "There's probably going to be some mistakes on the way, but I'm excited right now and happy to be back."