Thiago was substituted in the second half of the 3-0 win against Crystal Palace and assistant manager Pepijn Lijnders said he would miss Tuesday's League Cup game at Norwich City and Saturday's league clash at Brentford.
"I can't say now how long it will take but definitely not tomorrow and not at the weekend," Lijnders told the club's website.
"Let's get the scan results back and we have a proper diagnosis. A shame as well because he was taking the game by the hand, he was a proper playmaker in my opinion. A shame that we miss him."
Thiago is the second midfielder Liverpool have lost to injury in recent matches after Harvey Elliott needed surgery when he broke his ankle against Leeds United.
Right back Alexander-Arnold will miss the third round tie at Norwich. Lijnders said it was not COVID-19 and that the 22-year-old did not have symptoms.
The game also comes too soon for forward Robert Firmino who returns to training this week after a hamstring injury.
Lijnders said Liverpool would use the competition to blood youngsters, adding that 20-year-old midfielder Curtis Jones would definitely start.
"We want to compete with a certain style of football, we want to press and be offence and dominant and control the game," he told reporters.
"If you want to play like this you need fresh legs, fresh minds and energisers on the pitch and that's what we try and do. To understand team selection you need to understand our ways.
"For us it's an important competition despite what people are saying because we want to introduce new talent and we have this group of players who want to become proper legends, we really look forward to this."