The 53-year-old was due to have what he described as "simple and straightforward" surgery on Monday to relieve a "tightness of the arteries".
However, after extensive tests Allen was allowed to return home and will take first-team training on Thursday.
Barnet are 12th in the table, two points off the play-off places.
A statement on the club website said tests on Allen's condition are ongoing.
Barnet manager Martin Allen will take charge of Saturday's League Two match against Crewe after a surgeon decided not to operate on a minor heart issue.