The 22-year-old signed a two-year deal from National League rivals Chester after scoring 17 goals last season.
Alabi's contract with the Blues had expired but due to his age Tranmere must pay compensation for him.
Chester confirmed that "as of this time a satisfactory offer of compensation has not been received" and so the matter may have to go to a tribunal.
Rovers have a wealth of forward options on their books including Andy Cook, James Norwood, Connor Jennings and Andy Mangan.
Now with the latest addition of Alabi, Mellon is pleased with the variety of strikers. He told the club's website: “I’m delighted that James has agreed to join us.
“He is different to the attacking options we already have so it makes us flexible tactically and will enable us to play in different ways as and when we need to.
“James is a good age and his goalscoring record stands up for itself, so we have plenty of firepower from all the strikers at the club”.