Vardy has failed to score in his last 10 appearances for club and country -- his worst run since going 23 matches without scoring in 2014-15.
Van Persie, 33, racked up his century of caps last October after coming off the bench during Netherlands' 2-1 Euro 2016 qualifying win in Kazakhstan.
Thierry Henry has told AFP that Arsene Wenger is too "hands on" to give up his job at Arsenal and take charge of a national team.
Glenn insisted Southgate's next two results in temporary charge of the senior team will have little bearing on whether he gets the job full-time.
Sir Alex Ferguson says Scotland's World Cup qualifier against England at Wembley next month is a must-win game.