The People says agents are working behind the scenes to see if a deal can be struck to move the Cameroon hit-man to Old Trafford in exchange for Louis Saha and £5 million.
United manager Sir Alex Ferguson has been on the look-out for another striker for some time and has grown increasingly frustrated with Saha's constant injury problems.
The Frenchman will not play unless he feels 100 per cent fit and, while his ability is undoubted, his willingness to put his body on the line when needed is questionable.
Eto'o, meanwhile, has himself been plagued by injury this season and has not featured for Barca since tearing a tendon in his right thigh.
However, he is on target to return just before Christmas and would be ready to go by the time any deal could be sanctioned in January.
The striker will be on African Cup of Nations duty and would disappear for the back end of January and early February. But, long term, that would not bother Fergie.