The White House and State Department spoke glowingly of Blair's credentials on Wednesday but said there was nothing to announce yet. Blair steps down next Wednesday.
"Obviously Prime Minister Blair has been very active and deeply involved in Middle East peace issues throughout his prime ministership," White House spokeswoman Dana Perino said.
She said that Blair and Bush speak often. "It would not surprise me if they have talked about what Prime Minister Blair would like to do following the end of his term ... but we don't have anything to announce today," Perino said.
Assistant Secretary of State David Welch, the State Department's top diplomat for the Middle East, is in London and met with Blair this week.
Israeli Prime Minister Ehud Olmert expressed support for Blair playing a role in Mideast peace.
"Officials in the prime minister's office are aware of this idea and Prime Minister Olmert is very supportive of Prime Minister Blair and of his continuing involvement in the Middle East and the Peace Process," Israeli government spokeswoman Miri Eisin said.
State Department spokesman Sean McCormack said Blair "is a person of great abilities and obviously somebody with great interest in the region and has made great contributions to the region."
James Wolfensohn, a former president of the World Bank, stepped down in April as international Mideast envoy for the Quartet of peacemakers — the United States, European Union, United Nations and Russia. The position envisioned for Blair was said to be enhanced in contrast to Wolfensohn's role. Members of the Quartet may meet in Paris next week.
Blair's office refused comment.
"There is a lot of speculation about what the prime minster will do after June 27, but we are simply not going to comment," said a spokeswoman at his Downing Street office, on customary condition of anonymity in line with policy.
During his tenure, Wolfensohn helped negotiate several agreements between Israel and the Palestinians. They included transferring control of the Rafah border crossing in southern Gaza to the Palestinians, a deal to improve the flow of goods in and out of Gaza, and the purchase of dozens of greenhouses from Israeli settlers for Palestinian use.
These efforts, however, hit repeated obstacles, and in some cases, the agreements were not honored.
The Quartet has called on Hamas, the Islamic militant movement, to renounce violence, recognize Israel and honor past peace agreements. Hamas has rejected these demands.