Saturday's clash at Cardiff City Stadium will be Solskjaer's first Premier League game as a manager, and will also see Bellamy available for the first time in two months after hamstring and knee injuries.
The 34-year-old has already retired from Wales duty and had previously hinted he could hang up his boots for good at the end of the current campaign.
But Solskjaer hopes he will get the chance to work a while longer with the veteran forward.
He said: "We have spoken together obviously and my thoughts are to play as long as you can, if you can play football and you feel you can justify your performances.
"That is my job now and of course Craig wants to get performing and get back out there, so hopefully I will motivate him and inspire him to prolong his career.
"He has been training with us for a week and is looking really sharp to be fair. He will be involved tomorrow and is in the squad. He is back and available."
Having enjoyed a winning start as Bluebirds boss courtesy of last weekend's FA Cup victory at Newcastle, the meeting with West Ham comes with an altogether different pressure.
The Hammers have shipped 11 goals in cup defeats against Nottingham Forest and Manchester City in the past week, and sit 19th in the Premier League table having not registered a top-flight win since the end of November.
Cardiff are themselves just a point above the drop zone, and victory over the Hammers would give them some valuable breathing space with away trips to Manchester City and Manchester United looming.
But Solskjaer does not feel Saturday's meeting is a "must-win" encounter.
"There is never a must-win game at this part of the season, you cannot tell players you have to win," he said.
"If we get a performance like the Sunderland performance we can get a result. So performance is first and foremost, you cannot think about consequences all the time.
"We have to improve step by step and get the performances and play like we want to. It is nice to win but we can never say we have to win."
Solskjaer has already added to his squad with the acquisition of Magnus Wolff Eikrem from Heerenveen.
A move for Molde's Mats Moller Daehli is also in the offing, while Cardiff have also approached Hannover over striker Mame Biram Diouf.
But the Norwegian manager was coy when asked for an update on deals for the duo.
He said: "With respect to the other clubs, I don't think I can speak too much about them. There are other names as well, we will announce something if something happens."
But the Cardiff boss was delighted to have captured Eikrem who, like Daehli and Diouf, he worked with during his time at Manchester United.
He said: "The boy was very close to playing for Man Utd. He was there five years so it must have been a dream for him and hopefully I can fulfil it, but he needs to perform
"He is a player I have admired since he was a kid. The first time I saw him he was 12 and you could see he was hungry to get on the ball all the time and he likes to give players presents with his passing. His passing range is his strength, as is his vision.
"From that boy he has now outgrown me and he is bigger than me. He is getting towards being the Premier League midfielder we always thought he would be, now it's about performing on the pitch when he gets chances.
"I think he compliments the other players we have, I would never have signed him for the sake of it, he compliments what we have. The four or five central midfielders we have now have different attributes. You might say he has a little bit no-one else has."