The Republic of Ireland was holding out for a move to England with the likes of Aston Villa, Tottenham, Sunderland and Newcastle linked with moves.
His former boss Martin O'Neill was regarded as favourite to land his signature, but a deal has failed to materialise with Villa seemingly opting to look elsewhere.
Spartak first moved for McGeady earlier in the summer, but their bid of around £8million was rebuffed.
It is now believed they could be ready to do business at £10million, but McGeady has to make a decision and Celtic want a quick answer.
The Bhoys have their UEFA Champions League qualifying campaign on their minds and if McGeady is to leave, they want the deal doing this week.
Spartak, like Celtic, also want a decision to be made sooner rather than later, and they will hope McGeady opts for a move to the Russian capital.