The Jambos boss is on course to take charge of the Milton Keynes outfit after the two clubs agreed a compensation package, according to BBC Sport.
Former Hearts player Neilson was handed his first managerial job by the club in 2014, guiding them to the Scottish Championship title in his first season and securing third place in the top flight last term.
He will replace Karl Robinson at MK Dons, who was dismissed by the League One club following their poor start to the current campaign.
Former Liverpool midfielder Steven Gerrard was previously linked with the post before turning it down.
Neilson is expected to oversee his final Hearts match on Wednesday evening when the club play host to Rangers in the Scottish Premiership.