In a visit to UN headquarters in New York, Lone said: "The Americans have been so waffly about the election outcome, that this decision today by the president is a very welcome development.
"Free and fair elections are at the core of democratic practice, and we are greatly encouraged by the US determination to take the lead.''
Bush wants Rice to demand an immediate end to fighting in the East African nation.
Violence since the disputed December 27 elections has led to the deaths of more than 1,000 people and displaced 300,000.
Lone said the move showed the recognition that "international pressure is essential" to ensure a peaceful resolution.
Rice is expected to assist peace negotiations currently being mediated by Kofi Annan, the former UN secretary general.
Rice will talk to Mwai Kibaki, Kenya's president, Odinga, and possibly civil society leaders.
Bush and Rice begin a visit to Africa on Friday.
A spokesman for Annan said on Thursday that an agreement to rewrite the constitution had been made.
This includes a power-sharing deal between the parties.
Government negotiator Mutula Kilonzo said: "The two parties agreed to write a new constitution."
This was not confirmed by the opposition.
Lone said: "We should not be fooled by the current relative calm to believe that peace is around the corner.
"Already we can see that the government has consistently tried to undermine the negotiations on this core issue and even if Mr Annan succeeds in forging a settlement, the world will have to be very vigilant in insuring the government implements it.
"But the key issue was and is remains today: the stolen election and the need to share power to overcome the anger amongst Kenyans. Maximum pressure needs to be put on the government to insure that an agreement on power-sharing is reached quickly."
Talks have been adjourned until Monday, but an announcement on the deal is expected at 5pm (1400GMT) on Friday.