England left-back Cole, whose current deal is due to expire at the end of the season, is believed to have turned down the offer of a one-year extension at Stamford Bridge.
As a result, speculation this morning has claimed that the 31-year-old is now prepared to seek employment away from West London.
That has alerted Premier League rivals United, who would not have to part with a transfer fee to land the defender, according to the Metro.
However, they are likely to face competition from Paris Saint-Germain and Real Madrid.
Cole joined the Blues from London rivals Arsenal in 2006.