The 21-year-old winger has built a strong reputation for himself at Celtic, having been on loan at the Scottish club since January 2016.
The England Under-20 international has been a regular starter under Brendan Rodgers at Celtic Park.
He has gained valuable first-team experience away from the Etihad, which has attracted interest from other top-flight clubs.
West Ham have been tracking Roberts long term, having missed out on signing him from Fulham back in 2015.
Leicester are also keen, having looked to include him in a deal with City for Riyad Mahrez last month, while French outfit Nice are admirers too.