The United boss fears Real Madrid will hijack their move for Pogba after seeing Portugal star Andre Gomes reject the Bernabeu club for Barcelona.
But United are hopeful of signing the 23-year-old before the season's Community Shield curtain-raiser - next Sunday August 7 against Leicester at Wembley.
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Paul Pogba is wanted by Manchester United and Real Madrid
But the France international would prefer a move to Real, with boss Zinedine Zidane keen on signing him.
"I don't know if he will come," Zidane said. "Until 31 August anything can happen.
"Everyone is interested in Pogba. He is a great footballer and when you are Madrid you always want the best.
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"But now he is not a Real Madrid player, he is a Juventus player. I can't say anything more."
Mourinho is prepared to look elsewhere should United fail to land Pogba, with Chelsea midfielder Nemanja Matic also an option.
"We made a nucleus of four priorities, four positions, priorities," he said.
"To give a certain balance to the squad, to give a certain push in terms of quality, and the qualities that I need, that I want, as we know especially with the ones with more vision."