The 27-year-old was due to have a medical, and was seen attending a hospital in the city on Saturday morning.
"There's no smoke without fire and there's been a lot of smoke," said Celtic manager Brendan Rodgers at his club's media conference on Friday.
"Scott's a player I know well. I'd hope to sign a player before the weekend."
Villa boss Roberto di Matteo confirmed on Friday that Celtic have made an offer for Sinclair.
Celtic open their Scottish Premiership defence at Hearts on Sunday, with a League Cup meeting with Motherwell to follow on Wednesday.
Seven days later they face Israel's Hapoel Beer Sheva in the first leg of the play-off round of the Champions League.
With Patrick Roberts facing up to three weeks out with a hamstring problem and fellow winger Gary Mackay-Steven still recovering from an ankle operation, Rodgers is keen to sign Sinclair, who moved to Villa from Manchester City last summer on a four-year contract.
The pair have worked together twice - at Chelsea and Swansea City.
"I had him in at Chelsea when he was 16 years of age. I know him probably as well as anyone and I know his qualities," added Rodgers.
"But there are a number of players that we are looking at and all of them we hope can improve the squad.
"The squad will be better and stronger by the time we arrive at the next European games."
Roberts was withdrawn during Celtic's 2-1 win over Astana on Wednesday, with Rodgers saying the on-loan Manchester City player was "a risk" for the play-off round.
"He could miss the next two to three weeks," he said. "I'm disappointed for him since he was as at really good level of fitness and operating at a high level."
Aston Villa winger Scott Sinclair has arrived in Glasgow to complete his proposed move to Celtic.