The 29-year-old was a key player for the Cottagers last season but has lost his starting berth to Sascha Riether this term.
Kelly is set to leave the west Londoners in search of first-team football, with a permanent move to Reading close to completion.
Reading have struggled defensively this season and Brian McDermott is keen to bolster his defensive options
If Kelly secures his switch to the Madejski Stadium he will become McDermott's third signing of this month's transfer window following the arrivals of Daniel Carrico and Hope Akpan."We will try to get one or two players in," Jol said of his January transfer plans.
"One example is right-back as Stephen Kelly, who went for a medical at Reading today. It is close to completion.
"I expect him to join Reading permanently."
Republic of Ireland international Kelly has been at Craven Cottage since 2009, making 71 appearances in all competitions.