Michael Cohen and the publisher of the National Enquirer forged an alliance over the years, looking out for the interests of Donald Trump and each other. Now, federal investigators are examining those ties as part of a wide-ranging probe into Mr. Cohen’s personal business dealings and his self-described role as Mr. Trump’s fixer.

In previously unreported interactions, some of which are memorialized in emails now under review, Mr. Cohen mediated a dispute between Omarosa Manigault-Newman, who had been a star on Mr. Trump’s...