There had already been suggestions from multiple sources this week of an offer worth in the region of €18m.
According to Tuttomercatoweb, a representative for Lazio is now in Istanbul to whittle down the details.
The 26-year-old centre-forward is a Kosovo international and this season contributed 17 goals with seven assists in 36 competitive appearances for Fenerbahce.
Muriqi was only purchased from Rizespor a year ago for €3.5m, so this is an impressive profit.
Lazio are trying to bolster their squad with options for coach Simone Inzaghi after qualifying for the Champions League and after a quite disappointing Europa League this season.