City are also rumoured to have Street on their shortlist, with a loan move possible.
The 24-year-old has been at Crewe all his career, playing over 100 times since 1997.
A hamstring injury has restricted him to just four appearances this season, including two while on loan at Luton.
City are thought to be looking at a handful of midfielders at the moment, and Street could be one of those.