The 19-year-old is currently on the fringes of the first-team at Elland Road and competition for places in midfield has been intensified by the additions of Ilia Gruev and Glen Kamara.
According to The Athletic, Swansea have made a loan offer for the midfielder.
He has already reportedly been the subject of a loan bid from Fleetwood Town, as well as a permanent offer from French outfit Valenciennes.
Gyabi joined Leeds last year, making the switch from Manchester City during the same window that Kalvin Phillips went in the opposite direction.
The England under-20 international has made five senior appearances for Leeds, the most recent of which came in the 4-3 win over Ipswich Town.
Leeds have already shown a willingness to loan out young prospects in the current window, having sanctioned temporary moves for the likes of Sam Greenwood and Sonny Perkins.