The 33-year-old has been a free agent since leaving the Red Devils upon the expiration of his contract in July after 12 years at Old Trafford.
Man United opted against offering De Gea a new contract and instead spent around £47m on Andre Onana, who has since made a mixed start to life on the red side of Manchester.
De Gea kept 190 clean sheets in 545 appearances for Man United across all competitions and won five major trophies with the club including the Premier League, FA Cup, EFL Cup and Europa League.
The Spanish shot-stopper was also named in the PFA Team of the Year five times during his time with the Red Devils and he also won the Premier League Golden Glove twice, claiming last season's award with 17 clean sheets.
However, De Gea has struggled to find himself a new club despite being linked with a number of potential suitors, including a surprise return to Man United to provide short-term cover for Onana when he represents Cameroon at the Africa Cup of Nations next year.
Clubs in Saudi Arabia have also been credited with an interest in De Gea, while a return to his native Spain with Real Betis has also been mooted, with a recent report claiming that the goalkeeper has held talks with the La Liga club.
It has been suggested that Real Betis could move for De Gea to provide competition for third-choice goalkeeper Fran Vieites, who has been filling in between the sticks due to the absence of injured duo Claudio Bravo and Rui Silva.
Asked about the likelihood of Real Betis signing De Gea, Pellegrini told reporters via the Manchester Evening News: "If you ask me I would say I don't think so.
"I think what happened with Fran [Vieites] was more or less what we expected from him. He trains every day, he is a goalkeeper who has many abilities, he did not have the slightest responsibility for the goal, he was calm and for me it was a very good performance.
"We know how difficult it is to practically debut in professional football with everything that the outside environment entails."
Vieites, 24, made his La Liga debut as a substitute in a thumping 5-0 away defeat to reigning champions Barcelona in September, before starting his first top-flight match in a 1-1 draw with city rivals Sevilla last weekend.
Real Betis currently sit seventh in the La Liga table, one point behind sixth-placed Real Sociedad in the final European qualification spot and seven points adrift of Atletico Madrid in fourth.