The 19-year-old, who has established himself at Under-21 level for Villa, arrives at Prenton Park to bolster boss Ronnie Moore's goalkeeping options.
First-choice keeper Owain Fon Williams will be released by Wales to ease Rovers' goalkeeper crisis ahead of Sunday's trip to Bradford, with Jason Mooney (knee surgery) Sam Ramsbottom (broken ankle) ruled out.
Moore told Tranmere's official website: "Jason Mooney and Sam Ramsbottom are both unavailable through injury so it was important we brought in cover for Owain Fon Williams.
"Bradley is highly thought of at Aston Villa and did well for their Under-20 side in Europe last season.
"I'd like to thank Paul Lambert (Villa boss) for letting Bradley join us for a month."