That’s according to Gloucester Live, who claim the Blues face competition from Burton and Peterborough for the 23-year-old midfielder.
The report says that Posh boss Darren Ferguson was at Whaddon Road on Tuesday night to take in the Robins’ 3-0 win over Macclesfield in League Two.
Broom’s performance saw him named man of the match, with a goal and assist recorded in the convincing victory.
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That took his personal tally for the season to six goals and four assists as the League One trio weigh up a January move for the former Bristol Rovers man.
Broom has been playing in a No10 role for Cheltenham this season but played most of last season at right-wing-back.
He’s made 63 appearances for the Robins following his release by Rovers at the end of the 2017-18 season and signed a new two-year contract in the summer.
The Newport-born ace played 13 times for Eastleigh in the 2017-18 season during a loan spell at the Silverlake Stadium.