The Belgian started on the bench against former club Everton last Sunday as Mourinho opted for a front three of Rashford, Juan Mata and Anthony Martial.
Rashford was given the central position and, though he failed to score, his runs in behind created space for Uniteds midfield in what was an encouraging attacking display for the Red Devils.
Lukaku has failed to score in his last eight matches and missed a sitter after coming off the bench against the Toffees.
However, Lukaku is said to have reacted well to being dropped and Mourinho is ready to start him against the Cherries, according to the Mirror.
That means Rashford is likely to be dropped with Mata expected to keep his slot on the right-wing.
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Uniteds trip to the south coast is the first of three consecutive away matches with ties against Juventus and Manchester City to come.
Alexis Sanchez and Jesse Lingard both returned to the bench last weekend but are not expected to start against in-form Bournemouth.