Coker, who is from Mobile, Ala., backed up Heisman Trophy winner Jameis Winston this past season. The sophomore played sparingly and injured his knee late in the year.
Shortly after Florida State won the national title, the school released him from his scholarship.
Coker's high school coach, Steve Mask of St. Paul's Episcopal School in Mobile, said Coker spent the weekend in Tuscaloosa after Crimson Tide coach Nick Saban visited the quarterback at his home earlier in the week.
"I know he had a very good home visit last week with Coach Saban," Mask said. He added that Coker's decision could come soon.
Alabama is looking to replace Heisman finalist AJ McCarron, who also went to St. Paul's. McCarron started for three seasons and led the Tide to two national championships.
Coker is on target to graduate from Florida State in the spring and would be eligible to play next season. The 6-foot-5, 230-pound Coker completed 18 of 36 passes for 250 yards and an interception this season. He tore cartilage in his knee on Nov. 9 against Wake Forest and missed the rest of the season. He was still on crutches at the national championship game.
The only quarterback currently on Alabama's roster who has thrown a pass in college is Blake Sims, who will be a senior. Alec Morris will be a sophomore next season after getting a little game action as a redshirt freshman in 2013.