The final leg of Liverpool's tour saw them lose 4-2 to Leeds United in Chicago on Sunday, with Andoni Iraola's side playing the fixture at Soldier Field, home of Chicago Fire, and training at the MLS club's performance centre.
Van Dijk, however, did not travel with the squad after being granted an extended summer break following the Netherlands' World Cup campaign.
According to the Daily Mail, Fire officials had hoped to use the trip to informally pitch a future move to the 35-year-old by showing him around their state-of-the-art facilities, which were also used by Germany during the World Cup, but were left disappointed by his absence.
Van Dijk's contract at Anfield runs until next summer, and from January he will be free to agree a pre-contract with clubs outside England, opening the door to a free transfer at the end of the season. Liverpool are understood to have rejected any approaches for the defender this summer despite the risk of losing him for nothing in 12 months, given how central his leadership and experience remain to Iraola's backline.
It remains unclear whether the hierarchy will offer him fresh terms before then. Chicago Fire's interest comes shortly after the MLS side completed a marquee two-year deal for Robert Lewandowski, and La Gazzetta dello Sport has separately reported that AC Milan are also positioning themselves to move for Van Dijk on a free transfer next year, having accepted a deal was not realistic this summer.
Van Dijk is currently on wages of around £350,000 a week, a deal he signed just over a year ago, and has been training at the AXA Training Centre since last week, having been joined by the rest of the squad, along with Cody Gakpo, following their return from the US tour.
Liverpool's captain has been a mainstay of the club's defence since his £75m arrival from Southampton in January 2018, a fee that was a world record for a defender at the time, and has since won every major domestic and European honour available during his time at Anfield.
His importance has only grown as Iraola looks to blend experience with a younger backline this season, with the Dutchman's leadership seen internally as difficult to replace even as questions persist over his long-term future beyond next summer.