The Portuguese has penned a five-year deal with the Biancocelesti and will be in Rome tomorrow to undergo a medical.
The Aquile have beaten off competition from Sevilla for the services of the 25-year-old, who signed a five-year contract worth €750,000-per-season last night.
Eliseu had refused to extend his contract with Malaga beyond 2010 and the Primera Division club have allegedly accepted a bid of just €1 million from the Eternal City outfit.
The winger, who made his Portugal national team debut this month, is expected to fly to Rome tomorrow to undergo a medical before being unveiled to the press.
Meanwhile, Lazio president Claudio Lotito is continuing his Spanish shopping spree and has set his sights on Valencia playmaker Ever Banega to replace the departing Cristian Ledesma.
Banega, who will turn 21 next Monday, spent last season on loan at Atletico Madrid, who have not taken up their option to make the move permanent this summer.
Valencia are reportedly willing to sell the former Boca Juniors man and Lotito is hoping to get the player on loan with the right to sign him permanently next year.
Banega would reunite with compatriot Mauro Zarate in the Italian capital - the duo won the Under-20 World Cup with Argentina in 2007.
Vince Masiello, Goal.com