Snodgrass spent last season on loan at Aston Villa in the Championship and Pellegrini, on his appointment as Hammers boss, told the Scotland international to lose weight if he wished to show his commitment to the cause.
Pellegrini had admired Snodgrass in opposition, when the Chilean was at Manchester City and the Scotsman was at Norwich, and informed the midfielder his future at the London Stadium depended on him.
The Hammers boss said: “There were very few words that I spoke with him at the beginning of the season. I gave him some demands about his weight, about his work.
“I always found him a good player, just will depend on what he wants, if he had a spot here for the season.
“He’s a very professional player. He’s doing it very well because he had desire to be here.”
Pellegrini’s side play Cardiff on Tuesday, seeking to back up last Saturday’s 3-0 win at Newcastle, which followed a 4-0 home defeat to Manchester City.
Asked the reason for the transformation of results, Pellegrini said: “Manchester City is a top team that doesn’t allow you to make mistakes.
• READ MORE - Scotland begin Euro 2020 qualifying campaign with away double-header “We conceded two easy goals. You cannot concede those goals to those big teams.”
The Hammers boss insisted the priority was Cardiff, even though, on paper, there appears to be an opportunity to climb the table in the next few weeks.
Fixtures with Crystal Palace, Fulham, Watford, Southampton and Burnley follow before the end of December.
Pellegrini added: “If we have our minds focused on Cardiff, it will be an important advance. To win three more points is what we must do here at home.
“One of the most important mistakes you can do is to think that because a team is in a bad position in the table it will be an easy game.
“The best games we played here were against Chelsea and Manchester United.
“I hope we are conscious tomorrow about that. I hope we give maximum of our pace and quality to have three points more (against Cardiff) and after that to think of the next rival.”
Defenders Aaron Cresswell (hamstring) and Fabian Balbuena (knock) are doubts and will be assessed before kick-off on Tuesday and Pellegrini may rotate his options.
Cardiff boss Neil Warnock on Sunday celebrated his 70th birthday. Pellegrini, five years his junior, hopes to be a manager into his eighth decade, too.
He said: “I hope I will have the chance to continue not only to 70, more also.”