The 21-year-old has had previous loan spells at Grimsby, Scunthorpe and Stockport and has made just one first-team appearance for the Premier League club.
"I'm delighted to be able to bring Shaleum to Tranmere for the season," boss John Barnes told the club's official website.
"He is a talented young player who will improve our defence.
"I'd like to thank Manchester City manager Mark Hughes for allowing Shaleum to join us on loan."
Logan has been given permission to play in all three cup competitions for Tranmere by Hughes.