In the summer of 2016, Paul Manafort, Donald Trump’s campaign chairman, proclaimed Midwestern states like Michigan and Wisconsin to be critical to the campaign’s strategy.

Behind the scenes, he was actively trying to execute that strategy — sometimes in highly unorthodox ways.

According to three people close to the 2016 Trump campaign , Mr. Manafort sought to open a back channel to the A.F.L.-C.I.O., the nation’s pre-eminent labor federation, which typically backs Democrats for president and had endorsed Hillary Clinton.

These people said the Trump campaign had hoped the federation would scale back get-out-the-vote activity intended to help Mrs. Clinton win Michigan and Wisconsin. To that end, they said, Mr. Manafort enlisted an intermediary friendly with officials at the A.F.L.-C.I.O. to suggest a mutually beneficial relationship.