The Portuguese star bagged a stunning brace for Malaga in their Champions League tie against Anderlecht last night, as the La Liga side ran out 3-0 winners in Belgium.
The Spanish club have been plauged by financial troubles, and according to talkSPORT, United have long coveted the versatile left winger.
Malaga's difficulties almost saw Eliseu join Portuguese club Benfica in the summer, but the Andalusian side managed to hold on to their player.
Eliseu was signed by Malaga two years ago for just over £1.5m from Serie A club Lazio. It's the 29-year-old's second spell at the club, and the Portugal international has grabbed his chance with both hands.
His Champions League experience, and ability to play left-back as well as on the wing, makes him an attractive option for Sir Alex Ferguson.
Manchester United fullback Patrice Evra has come in for criticism in recent weeks, and his place is under threat following the summer purchase of Dutch defender Alexander Buttner.
And talkSPORT speculate Manchester United could move for Eliseu in January, as they line up a long-term replacement for the 38-year-old Ryan Giggs.