The former Montpellier man has made just three Premier League appearances this term, all from the bench, and has yet to find the net in the league.
Rashford, 18, made his United bow back in February, scoring two goals in a Europa League win against Midtjylland.
Eddie Howe has said he is "committed" to Bournemouth but reiterated his belief that managing England is "the ultimate job."
Southgate, the England under-21 coach, has been temporarily promoted to the senior role after the departure of Sam Allardyce.
Allardyce was forced to step down as England boss on Tuesday after just 67 days in charge after sting operation.
The Serbian football federation has appealed to the Court of Arbitration for Sport against Kosovo being accepted as a member of UEFA.