The offer from the Saudi club is the only one Messi has received so far, the source added. Argentine media have reported that the offer is worth around $400 million a year.
Al-Hilal did not immediately respond to a request for comment outside of normal business hours.
Messi has also been linked in the media with a move back to boyhood club Barcelona, with the United States' Major League Soccer club Inter Miami also being touted as a possible destination.
He was suspended by his current club Paris St Germain at the start of the week after making an unauthorised trip to Saudi Arabia, French sports daily L'Equipe reported on Tuesday, adding that PSG had decided against exercising an option to renew his contract for a third season.
Messi is a Saudi ambassador for tourism and his long-time rival, Cristiano Ronaldo, signed for Saudi club Al-Nassr in December in a deal reported to be worth around $220 million per year.
A second source close to Messi told Reuters the suspension was imposed after the forward told PSG he would not stay in Paris following this season and felt the club lacked a project.
The source added that PSG had no training scheduled on Monday and the decision to hold a session was taken only after Messi was already in Saudi Arabia.
A source close to PSG denied this, telling Reuters Messi travelled without permission from the French club and despite there being a training session scheduled on Monday.
The source did not comment on whether Messi had already told PSG he intended to leave at the end of the season.
