The Ivory Coast international has praised the new manager, who has overseen an upturn in form and given the defender starts, and wants to be a part of the team's future
The Ivory Coast international, who looked set to leave The Hawthorns in the January transfer window, has been impressive of late after playing seven consecutive matches for the Baggies and suffering only one defeat.
"Yes, I did think I might need to leave," Meite told reporters.
"It was hard for me but I've tried to be professional because I'm a footballer. This is life.
"I had a chance to leave in January. There were a few things but I said I preferred to stay here and would then see in the summer.
"Football, you can be the hero one day and on the floor the next day. I was on the floor but now I'm playing I am okay.”
The 30-year-old has no doubt West Brom’s recent return to form - and all but secured survival in the Premier League - is down to former Liverpool and Fulham boss Hodgson, who took over The Hawthorns hotseat in February following the sacking of Roberto Di Matteo.
"I want to stay because we have a new manager and he can do a good job here," Meite told the club's official website.
"He is experienced, he is strong. He knows what we need to do.
"Our shape is stronger and different and he talks with everyone in the squad and he's there all day and all of the night for us.
"He's good for us and good for the club. He did a good job at Fulham, taking them to the Europa League. He can do that here."
Before Hodgson’s arrival, Meite had not made a single appearance in the league this season, but after impressing the new boss the defender hopes to be offered a 12-month contract extension on his current deal.
He added: "I am happy here and yes, of course, I would like to stay.
"I know I have had something to prove to the fans, to the club and to myself.
"This is now my time to try and do it. I still have a few years left. I am confident of playing some more."