O'Neill signed Bellamy on loan from Newcastle when he was at Celtic two years ago but is not prepared to pay significantly over the odds for the Wales international.
Martin O'Neill Fears Toon In Transfer Market
O'Neill's summer targets depend on club owner Randy Lerner but the fact other clubs, including West Ham and Newcastle, have financial muscle could make his task more difficult.
O'Neill: "It won't be easy with the big clubs having so much more spending power, but that's the beauty of it. I'm not sure it's the case we're all going for the same players. But one thing is for sure, we're going to to try to improve and we have our own priorities.
"West Ham made their intentions very clear in January and they have made them clear now so I expect them to be signing very big players in the transfer market in the summer.
"They have got renewed life from staying up and there's no doubt that, if they were already making inroads in January, they are certainly going to go for it now as well.
"Newcastle will be giving it a big go now as well - as they always do."