The striker completed a move from Liverpool in a deal reported to be worth £19 million ($33.4 million), having struggled for first-team opportunities under Jurgen Klopp.
The England international had appeared likely to join Crystal Palace on loan until the move fell through, with fitness concerns rumoured to be the reason after a medical at the London club.
Howe says Solanke could miss the rest of January with a hamstring problem, which Bournemouth knew about before completing the deal.
"Dominic will miss most of January," Howe said after his side's 3-1 FA Cup third-round loss to Brighton and Hove Albion.
"He's got a bit of a hamstring issue so it will be more February before we're likely to see him in action.
"But we knew about that because he's been through all the medicals and checks."