Albion are believed to have paid Boca around £600,000 to secure his early release.
He is currently with Argentina’s under-23s in Colombia, where he is taking part in the Olympic Games qualifying tournament and has scored two goals.
Mac Allister will link up with Albion after the competition finishes on 9th February, and is expected to be available after the one-week break.
Albion head coach Graham Potter said, “We’re delighted to have Alexis with us for the rest of this season, having now been granted his work permit. He will give us another attacking option from midfield. He is creative and capable of scoring goals.
“Alexis is a really exciting talent, who will bring an added dimension to our squad, and I have no doubt our supporters will be looking forward to seeing him play for the club.
“I have watched his progress with great interest in the past year, playing top-level football in South America and making his senior international debut for Argentina. I am now looking forward to working with him day to day.”
Albion’s technical director Dan Ashworth was in Buenos Aries last week to negotiate Mac Allister’s return to the club following receipt of a work permit.
He added, “Alexis’s rapid progress in the past year, both at domestic and international level, enabled us to secure his work permit during this window, and we have accelerated our plans to bring him in to our first team squad.
“We are grateful to Boca Juniors for releasing him early from the season-long agreement. He will finish his international duty with Argentina, and join up with the squad after our match against Watford.”
The 24-year-old attacking midfielder has been with Boca since the start of the season, making 19 appearances across the Argentina Super League and Copa Libertadores.
After joining Albion from Argentinos Juniors last January on a four-and-a-half year contract, Alexis returned to the Buenos Aires club on loan and scored two goals in his ten appearances.
He then joined city rivals Boca Juniors and was called-up by the Argentina national team for the first time in August 2019 and made his international debut as a second-half substitute in the 0-0 draw against Chile in Los Angeles.