Foster missed two field goal attempts and had a third blocked in the Crimson Tide's 34-28 loss to Auburn on Nov. 30.
He posted a photo of the letter on his Twitter and Instagram accounts Wednesday.
"Dear Cade (#43),
"Life has its setbacks. I know!
"However you will be a stronger human with time.
"I wish you all the best —
Bush was the 43rd President and Foster wears No. 43.
"Framing this," the kicker wrote on the Instagram caption
The letter was first reported by al.com.
No. 3 Alabama will play Oklahoma in the Sugar Bowl, but had been trying for a third straight national title.
Foster received threats on social media in the aftermath of the loss to Auburn. But teammates and Alabama fans swiftly rallied around him, with thousands joining a Facebook fan page "Bama fans supporting Cade Foster" by the following day.
Foster had made 11 of 12 field goals on the season before the game with Auburn.
He is a senior.