The Clarets central defender is a former academy graduate having joined United at the age of 10, and is being touted for a potential return to Old Trafford.
Rumours have linked the Reds with a bid but it would appear they are deadly serious after paying Turf Moor a visit on Saturday for the clash with Manchester City.
Stockport-born Keane did not have the best of games, failing to prevent City striker Sergio Aguero bagging a victory brace for Pep Guardiola's team.
England caretaker boss Gareth Southgate was also present to watch the 23-year-old go through his paces. United are not the only interested party, with both Leicester City and Chelsea also said to be in the hunt.