The move for Engel was made retroactive to Friday. Engel also began the season on the IL with a hamstring injury.
The 29-year-old Engel is batting .248 with seven homers and 18 RBIs in 33 games. He also has been dealing with a groin issue.
“The expectation is that 10 days from now, he’ll be active and ready to play,” La Russa said.
Wright, 31, is looking for his first big league appearance since 2019. He signed a minor league deal with the White Sox on Dec. 24.
The 6-foot-6 Wright is 7-5 with a 3.40 ERA in 16 starts with Charlotte this year. He is 10-12 with a 6.00 ERA in 110 major league games with Baltimore and Seattle.
“There will be no hesitation to use him where we need innings,” La Russa said.
Burr has made 19 appearances with Chicago this year, going 2-1 with a 2.95 ERA.
Foster pitched the ninth inning in Sunday’s 5-3 loss to the New York Yankees, surrendering Luke Voit’s two-run homer. After making an impressive major league debut in 2020, Foster is 2-1 with a 5.67 ERA in 31 games with the White Sox this season.