The center-right Conservatives won the most seats in the 650-seat House of Commons, comfortably ahead of the ruling center-left Labour Party but not in overall control. The centrist Liberal Democrats came out a distant third.
Prime Minister Gordon Brown remained in office in a caretaker role pending the emergence of a new government, in accordance with British constitutional convention.
He said the Conservatives and Liberal Democrats had every right to try to do a deal first but was ready to talk to the Liberal Democrats about a possible agreement if discussions failed.
Thursday's poll, marred by scenes of anger as hundreds of voters were unable to cast their ballots due to administrative problems, left none of the three big parties satisfied.
The Conservatives came tantalizingly close to an outright win but were left at the mercy of smaller parties. Ahead in the polls by double digits for most of last year, they saw their
lead shrink in recent months despite voter fatigue with Labour.
For Labour, hopes of extending their 13 years in office looked tenuous, while a predicted Liberal Democrats surge did not happen.
With results in 649 out of the 650 parliamentary constituencies declared, the Conservatives had won 306 seats, followed by Labour on 258 and the Liberal Democrats on 57.
"A strong basis for a strong government"
Conservative leader David Cameron said Britain urgently needed a strong government to reassure jittery markets that it was serious about tackling the country's record budget deficit, which exceeds 11 percent of national output.
"No government will be in the national interest unless it deals with the biggest threat to our national interest, and that is the deficit. We remain completely convinced that starting to deal with the deficit this year is essential," Cameron said.
The Conservatives were in power for most of the 20th century but they lost three successive elections to Labour from 1997. Cameron is under intense pressure to lead them back into office.
He said one possibility was a minority Conservative government seeking support from other parties issue by issue, but he would also offer the Liberal Democrats a more formal agreement, which could include coalition government. This is extremely rare in British politics.
"I want to make a big, open and comprehensive offer to the Liberal Democrats," he said. "I think we have a strong basis for a strong government."
Possible areas of agreement included reforming tax and electoral systems and reversing a planned increase in payroll tax, Cameron said, signaling that the Conservatives would be less open to compromise on European and defense issues.
Cameron has said he would make deeper and faster spending cuts than Labour or the Liberal Democrats, who have both warned this would harm a fragile recovery.
His statement went some way to calming investor fears of political deadlock and British government bonds briefly erased their earlier losses after he spoke -- the latest lurch on a volatile day.
Sterling also recovered partly from an earlier fall.
Brown holds out carrot to Liberal Democrats
The Liberal Democrats, whose leader Nick Clegg had stated earlier that the Conservatives should have the first shot at forming a government, said after Cameron's statement that it was time for a "breather" and Clegg would not speak publicly again on Friday.
Clegg and Cameron spoke later on the phone but no deal is expected imminently. The Lib Democrats are expected to discuss any proposals on a deal internally on Saturday.
Appearing earlier outside his Downing Street residence, Brown said that if the two opposition parties could not clinch a deal, he stood ready to talk to the Liberal Democrats. He dangled an offer of electoral reform, a key Lib Dem demand.
A coalition with the Liberal Democrats is Brown's last hope of holding onto power.
The BBC calculated the Conservatives had taken 36 percent of the overall vote, Labour 29 percent and the Liberal Democrats 23 percent.
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