The 21-year-old came through the youth ranks at the Gunners before joining Swansea in 2015 and spent much of last season on loan at Barnet.
He made 23 appearances for the Bees and joins the League Two Imps after his release by the Swans.
Vickers previously played under Imps management duo Danny and Nicky Cowley at Concord Rangers in 2014-15.
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Lincoln City have signed former Arsenal and Swansea City goalkeeper Josh Vickers on a two-year contract.