They face Anderlecht in Belgium on Thursday before a home game against Chelsea on Sunday followed by the second leg of the quarter-final at Old Trafford next week.
"When it is still mathematically possible to finish in the top four I think if we play against Chelsea with our second team you would kill me. The football country would kill me," Mourinho told British media.
"There is no reason not to try while mathematically possible. If on a certain moment of the season we have no chances to get top four and we are still in the Europa League then nobody can criticise."
Mourinho is hoping that key players do not join United's injury list as they try to win the Europa League and break into the top four simultaneously.
"Hope no more injuries, no more suspensions and hoping the players keep showing amazing character to keep going and going. Until not possible we are going to fight for the top four," he added.
United will be without captain Wayne Rooney against Anderlecht while goalkeeper David de Gea faces a late fitness test.
(Reporting by Aditi Prakash in Bengaluru)
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