Striker Beardsley was forced off shortly after giving Boro a 38th-minute lead against Rovers with his eighth goal of the season. Captain Mark Roberts made it 2-0 late on before the visitors grabbed a consolation goal from the penalty spot.
Beardsley sustained the injury in finding the back of the net and the 27-year-old is set for a scan to determine the damage caused.
Meanwhile, young striker Michael Thalassitis has earned a call-up to a Cyprus Under-19 training camp.
Thalassitis scored 17 goals in 26 games for Boro's Under-18 side last season and after making his first team debut this year at MK Dons joined Boreham Wood on loan in October.
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