The Latics had hoped to unveil the Spaniard in the JJB Stadium hot-seat last Thursday after agreeing a fee with the Championship club to speak with their manager at the start of the week.
Personal terms between Martinez and Wigan were agreed ahead of a press conference, but his confirmation had to be put on ice when Swansea demanded further compensation for several members of his backroom staff.
However, following further negotiations over the weekend and throughout Monday, the two clubs have now reached a settlement which means Martinez is now free to take the reins at the Premier League side.
The disagreement between the two clubs surrounded the futures of assistant manager Graeme Jones, chief scout Kevin Reeves, goalkeeping coach Inaki Bergara and masseur Oscar Brau.
Barnsley have allowed goalkeeper Heinz Muller to join Bundesliga side Mainz in a £600,000 deal.
Muller joined the Tykes from Lillestrom two years ago for £150,000 and made 69 appearances during his time with the South Yorkshire club.