The Frenchman, 23, joined the south-coast outfit from Le Havre in 2016 for a reported £5.7million and has made 71 appearances for the first team, mostly as a substitute.
Mousset has scored five goals for the Cherries, netting in his only top-flight start of last season in a 5-1 loss at Arsenal in February.
The Blades have been linked with a number of striking targets, including Swansea City frontman Oli McBurnie and Brentford's Neal Maupay.
United saw a £15m bid for Leeds-born McBurnie rejected earlier this week, with Bradford City entitled to a 15 per cent sell-on fee as part of any prospective deal after selling the former Bantams academy forward to the Swans in 2015.
So far this summer, the Bramall Lane outfit have broken their transfer record twice to bring in Preston forward Callum Robinson and QPR midfielder Luke Freeman.