The 28-year-old will move to the Kilmac Stadium "in the near future" once all the necessary paperwork is finalised.
Chapman has been a free agent since November after leaving Inter Miami, where he played 25 times under former England international Phil Neville in the MLS. Prior to moving to Miami - owned by David Beckham - in 2019, he was with Toronto.
The midfielder has played three times for Canada, most recently in 2020.
He becomes James McPake's first signing of the current transfer window.
The Dark Blues’ boss is also keen to reinforce his defensive options, and has run the rule over former Dundee United centre-back Paul Watson, who has been without a club since being released by Dunfermline in October.
Out-of-favour Celtic defender Osaze Urhoghide has also been linked with a move to Dens Park with reports suggesting discussions have already taken place over a possible loan move until the end of the season.
Dundee already have Leigh Griffiths on loan from Celtic, although there is the potential for that deal to be terminated this month.
Urhoghide signed for Celtic from Sheffield Wednesday during the summer, but has struggled to break into the first team under manager Ange Postecoglou.
The Glasgow outfit are keen for the 21-year-old from the Netherlands to get more minutes under his belt and are open to a loan move in this window.
Clubs in England are also keen to land Urhoghide, with proposals coming in from Championship and League One.