Striker Best, who has made two appearances for the Republic's Under-21 side, is one of six uncapped players in Giovanni Trapattoni's 25-man party for the game at Fulham's Craven Cottage on May 29.
Westwood will again compete with vastly experienced goalkeepers Shay Given and Dean Kiely in preparation for the World Cup Group Eight qualifier against Bulgaria in Sofia the following weekend.
Coventry chairman Ray Ranson believes the call-ups are vindication of the club's recruiting policy, and he told City's official website: "Leon and Keiren are great examples of types of players the club needs to take us on to the next level.
"They are both young and are hungry to improve and we want them to do that here. Both are clearly very much are part of the future for the Republic of Ireland and it is testament to the way they have acquitted themselves in the past 12 months for us that they are in the full squad.
"We are all delighted for them and look forward to watching their international progress because their personal success reflects well on the club."