Reid did not attend the post-match press conference on Saturday, in the wake of reports that he could quit just days after being given a vote of confidence by the Leeds board following six months in charge.
Having watched Leeds win a vital 2-1 victory over Blackburn that lifted his team out of the bottom three, Reid is now keen to press McKenzie in guaranteeing some time to make further progress.
Leeds have reportedly identified Nottingham Forest manager Paul Hart as Reid's successor should he go, but stopped short of releasing Reid last week after McKenzie conducted a straw poll of fans' representatives.
If Reid did quit, Leeds would not be obliged to pay up his £850k one-year rolling contract but would be in turmoil again before a run of tough matches in October.
McKenzie, who has emerged badly from the midweek farce, is due in Japan this week and Reid wants assurances before the visit of champions Manchester United on October 18th.