The 30-year-old forward becomes West Ham's first signing of the January transfer window.
Ings spent one-and-a-half seasons at Villa Park, scoring 13 goals in 48 appearances.
Ings also had spells at Southampton, Liverpool and Burnley, scoring 55 goals in 140 matches across his time at the three clubs.
If West Ham submit the required completed documents by midday Friday, Ings could make his debut the next day when they welcome Everton in a crucial relegation battle.
"It's important I settle in as quickly as I can do – and do the important stuff on the pitch for West Ham," Ings told the club's official website.
"We’ve got some massive games ahead of us – and I've come here to try and score as many goals as I can to help the team to get some positive results.
"I'm really looking forward to getting started."