The report claims former-United coach Rene Meulensteen, now in charge of Fulham, is keen to bring the talented young pairing to Craven Cottage.
Both starlets are tipped for big things at Old Trafford, but Meulensteen hopes his previous links with United will help him secure the reinforcements he needs to increase Fulham's chances of Premier League survival.
Offers have been made for both United players, and Fulham are now waiting to hear back. They reportedly remain hopeful that a deal can be reached by Friday's deadline.
Cole, a 20-year-old striker, is yet to feature for United in the Premier League. He has three caps for the England under-20s, and made his senior debut for United in September 2011.
Tunnicliffe is a 21-year-old midfielder who has enjoyed three loan spells - at Peterborough, Barnsley and most recently Ipswich Town.
Like Cole, he is also yet to feature for United in the Premier League.