That player, who will be identified soon, is believed to be cornerback Jamar Fletcher, the former first-round pick who has been a big disappointment.
The condition for the draft pick is that Boston make the Dolphins' 53-man roster.
The Dolphins already agreed on a new contract for Boston to replace the seven-year, $47 million deal he signed with the Chargers last offseason.
The new deal is believed to be heavy on incentives and includes little up-front money.
Boston, the eighth overall pick in the 1999 draft, led San Diego with 70 catches last season but the Chargers decided to let him go after just one season because he wasn't a good fit, according to GM A.J. Smith.
A press conference is expected to be held shortly after the deal becomes official.
In other news:
-- The Dolphins have made official the signing of offensive tackle Damion McIntosh. He signed a six-year contract, and it's believed to be worth $23 million.
-- The Dolphins waived safety Antwoine Sanders, who had been signed as a "street" free agent on Jan. 8.