Burge had a period of absence from football, with test results in the aftermath of testing positive for COVID-19 raising some concerns.
The 29-year-old therefore went through a period of mandated rest, keeping his heart rate below 100.
Neil says further tests have cleared the goalkeeper to take tentative steps towards a return, though the head coach added that they would be cautious in their approach.
The stunning £975m market values of Championship squads this season and where Sunderland rank alongside Sheffield United, Middlesbrough and co
Sunderland favourites to sign Nathan Broadhead with new Everton contract unlikely
Sunderland AFC transfer gossip: Aiden McGeady urges Celtic's James McCarthy to join Alex Neil on Wearside
Transfer news: Leeds United set to sign Sunderland 'target' with ex-Wigan Athletic and Everton man set for move
Sunderland AFC transfer news: Cats miss out on £2.5m 'target' as ex-defender makes Switzerland move
It is nevertheless a major boost for the player to be taking significant steps towards a comeback further down the line.
“We've had good news on Burgey,” Neil said.
“To be honest, the only reason I haven't mentioned it is that I don't want to put any pressure on him at all.
“When you're talking about vital organs, the last thing you want to do is start talking about people coming back. His immediate health is the most important thing.
“He's in a much better place now. His test results came back relatively normal, and we're just going to slowly, gradually build up doing a little bit more work at a time and make sure that we don't over-exert him.
“He's been in and about it, he'll be on the grass today, but he'll be doing limited amounts [to begin with] because we need to make sure that we look after him.”
Burge’s last Sunderland appearance came against Arsenal in the Carabao Cup quarter final last December.