It follows the announcement of a series of London Underground strikes over job losses faced by Metronet staff. Metronet planned to invest £17 billion in the Tube but went into administration due to serious financial problems.
TSSA general secretary Gerry Doherty said: "Returning Tube maintenance to TfL could ultimately deliver a stable, economic and efficient structure that is better able to meet the needs of Londoners than that delivered by the failing private sector under the PPP arrangements."
RMT leader Bob Crow appeared to welcome the move and said: "It is the RMT's view that tube maintenance should be back in the public sector under the direct control of London Underground. If this is to be a serious bid it will obviously be a welcome step in the right direction."
A TfL spokesman said it would submit its formal application to run Metronet's Tube maintenance service at the end of September.