Manchester United are reportedly aiming to sign five new players before the end of July.
The Denmark international has allegedly verbally agreed a three-year deal with the Red Devils, and he could be presented as the club's second signing of the summer ahead of their pre-season tour.
According to the Manchester Evening News, Man United are looking to complete five signings by the time that their tour of Thailand and Australia has come to a close.
The Man United squad will fly out to Bangkok on Friday, ahead of their pre-season opener against bitter rivals Liverpool on July 12.
The 20-time English champions will then take on Melbourne Victory, Crystal Palace and Aston Villa in Australia before returning to Europe to face Atletico Madrid and Rayo Vallecano ahead of their Premier League opener.
The Red Devils will face Atletico and Rayo on back-to-back days at the end of the month, with the clash against the latter set to represent Ten Hag's first home game in charge.
Man United will then begin their 2022-23 Premier League campaign against Brighton & Hove Albion on August 7, while Brentford will be their first away opponents of the new season on August 13.
A deal for Ajax defender Martinez is said to be heading in the right direction, but Man United might be priced out a deal for Antony, who is believed to be valued at £80m by his Amsterdam club.
Meanwhile, De Jong's switch has been thrown into doubt due to recent comments from Barcelona president Joan Laporta, who has claimed that the club are not looking to sell the midfielder this summer.
"There are clubs that want him, not just United. We have no intention of selling him, he wants to stay. Frenkie is considered as one of the best midfielders in the world by all the experts. We are happy to have him," Laporta told reporters.
"I'm going to do everything I can in my power so that Frenkie stays here, but there's also a salary issue and that would have to be adjusted."
Lee Grant, Jesse Lingard, Juan Mata, Paul Pogba, Nemanja Matic and Edinson Cavani have all left Old Trafford this summer, while Eric Bailly, Phil Jones, Andreas Pereira, Axel Tuanzebe, Anthony Martial and Alex Telles could potentially depart before the end of the transfer window.