Winston visited the Chargers three weeks ago and has been linked to the Miami Dolphins, but the "door remains open" in San Diego, according to the source. Winston, who started 16 games with the Chiefs last season, would loom as a right-side replacement for tackle Jeromey Clary. Gaither's fill-in figures to come in the 2013 NFL Draft, possibly with the Chargers' No. 11 pick.
Winston has taken his time in free agency. His agent said this week he would "really like" to play for the Dolphins after almost signing with the team last offseason. And another report on Thursday said the Philadelphia Eagles -- one of the first teams to show interest in Winston -- recently checked back in with his agent.
ProFootballFocus ranked Winston as the ninth-best right tackle in 2012, and he won't be out of work for long.
Follow Marc Sessler on Twitter @MarcSesslerNFL.