United's executive vice-chairman Ed Woodward is finalising a one-year deal with the player's representatives and the 34-year-old could join before Euro 2016, Sky sources said.
Sweden's first Group E match against Ireland is on June 13.
The former Malmo, Ajax Amsterdam, Juventus, Inter Milan, Barcelona and AC Milan striker, whose Paris St Germain contract is about to expire, wants to add the Premier League title to a trophy cabinet that already includes the Dutch, Serie A, La Liga and Ligue 1 crowns.
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The prolific Ibrahimovic scored 11 of Sweden's 19 goals in Euro qualifying including three out of four in a playoff win over Denmark that booked their place at the finals in France.
The tall Swede finished his career at PSG by helping the club to their second successive treble with two goals in a 4-2 victory over Olympique de Marseille in the French Cup final.
Ibrahimovic had already celebrated his last PSG game at the Parc des Princes against Nantes by breaking the club's scoring record in a top-flight season with a double, taking his tally to 38 goals, one more than Carlos Bianchi in the 1977-78 season.
(Writing by Ken Ferris; Editing by Tony Jimenez)
