Salah was replaced by Diogo Jota in the 33rd minute at Wembley Stadium after feeling discomfort in his groin.
Van Dijk, meanwhile, did not feature in extra-time after reporting what he later described as "a little twinge" in his knee.
However, it is understood that neither player has been sent to undergo a scan on Sunday such is Liverpool's optimism regarding their condition.
Only in the event that either Salah or Van Dijk report continued pain on Monday will further investigation be undertaken.
And that outcome is currently considered unlikely by the club's medical staff.
— Anfield Watch (@AnfieldWatch) May 14, 2022
Liverpool showed their preference for caution in withdrawing both at Wembley and will take a safety-first approach again with a Champions League final on the horizon.
However, the opportunity to pile further pressure on Manchester City ahead of the final day of the season has emerged following the champions' 2-2 draw with West Ham.
A win over Southampton would move the Reds within a point of the league leaders, who host Steven Gerrard's Aston Villa next Sunday.