The 35-year-old was photographed outside Ross Hall Hospital in Glasgow ahead of a medical on Saturday morning.
The Ivory Coast international could now sign for Celtic before they fly out to Kazakhstan on Monday for their Champions League qualifier against Astana.
The former Manchester City and Arsenal centre-back will be the second signing made by his one-time Anfield boss Brendan Rodgers since he took over as manager.
Two Celtic central defenders, Jozo Simunovic and Dedryck Boyata, are injured while Efe Ambrose was dropped for the second leg of their triumph over Lincoln Red Imps after his mistake cost the Scottish champions a losing goal in Gibraltar.
Celtic have until the eve of Wednesday's third qualifying round first leg to make one wildcard signing that could feature against Astana.
Toure said he hoped the deal was tied up before Celtic's trip to Astana.
He told Sky Sports News: "It is a privilege to be linked with such a big club and I am just looking forward to it.
"I am a ready for a new challenge.
"I'm here to try to be ready and help the team as much as I can.
"The Champions League is a great competition and I always look forward to it.
"It's really important, the game coming, to qualify and we will give everything we have got to qualify.
"I plan to work hard every day in training and help the team to get better, that's all I know."
Source : PA
Source: PA