The Manchester Evening News says United had beaten City to sign the 16-year-old defender, who had a trial at Old Trafford in August last year.
Bejger was not included in United's academy intake last month as they awaited clearance to complete his move from Lech Poznan. United initially agreed the deal for Bejger in January.
Poland Under-16 international Bejger made his youth debut for the national team against Scotland in September and captained the U16s against Moldova earlier this year. Bejger is primarily a centre-back but can also play at right-back.