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Julian Assange is at the Ecuadorean Embassy in London.Credit John Stillwell/Agence France-Presse — Getty Images

LONDON — Time is running out for Swedish prosecutors investigating accusations of sexual assault against Julian Assange, the founder of WikiLeaks, who has been holed up in the Ecuadorean Embassy in London since he walked into the building in 2012 and requested asylum.

Swedish officials said Wednesday that three of the four claims against him may never be investigated because the time Swedish law allows for the authorities to do so is about to expire.

It is the latest twist in a story that has often seemed like a grueling diplomatic and legal war of attrition. But it is not the final chapter. The fourth and most serious allegation against Mr. Assange, of rape, can remain under investigation for a further five years, and after some fractious exchanges, Swedish and Ecuadorean diplomats have scheduled more talks to try to end the impasse.

British police officers remain on a 24-hour watch outside the Ecuadorean Embassy in the Knightsbridge district of London, unable to enter the building because of diplomatic protocol but ready to arrest Mr. Assange if he leaves.

The police surveillance has already cost British taxpayers 9.2 million pounds (about $14 million) from June 2012 to April 2015.

No formal charges have been filed against Mr. Assange, who denies the allegations, originally made in 2010. But he has refused to go to Sweden to face an investigation because he says he fears he would then be extradited to the United States, where he could face a trial over the publication of leaked diplomatic cables on WikiLeaks.

Swedish officials say that there has been no extradition request from the United States and that his fears are “hypothetical.”

In March, it appeared that a breakthrough was imminent when Marianne Ny, Sweden’s director of public prosecution, who had previously insisted that Mr. Assange be interviewed in Sweden, said prosecutors were prepared to travel to Britain to question him.

Since then, Sweden and Ecuador have sparred over a legal agreement needed before questioning can take place. Though Sweden says it sent a prosecutor to London in June, the interview never happened, and each side has blamed the other.

Sweden’s Justice Ministry said that it could not accept, on legal grounds, a request by Ecuador to recognize its decision to grant Mr. Assange asylum, and so guarantee that Sweden would not eventually extradite him to the United States.

On Monday, the Ecuadorean Embassy in London issued a statement saying, “At no point has the Republic of Ecuador asked the Kingdom of Sweden to grant Mr. Assange asylum.”

It also said, “On no occasion has any representative of the Kingdom of Sweden presented themselves at the embassy in relation to the Assange matter.”

Karin Rosander, director of communications for the Swedish prosecution authority, said an assistant prosecutor, Ingrid Isgren, traveled to London on June 16 and 17, ready to begin the questioning, but had not gone to the Ecuadorean Embassy because it was clear that the legal formalities had not been completed.

But talks on a broader mutual assistance deal are scheduled to go ahead, said Cecilia Riddselius, a senior Justice Ministry official.

On Thursday, the statute of limitations will expire on one allegation of sexual molestation and one of unlawful coercion, Ms. Rosander said. Next Tuesday, another allegation of sexual molestation will reach its expiration date.

Ms. Riddselius said that in practical terms, it would be impossible to proceed given the time required for normal legal procedures. However, the allegation of rape could be pursued until August 2020. Under Swedish law, Mr. Assange could face up to four years in prison if convicted of the type of rape allegation made against him, the least serious in that category of offense.

Per Samuelson, a lawyer for Mr. Assange, said that technically, it might still be possible for prosecutors to pursue all of the allegations, even those whose statutes of limitations are about to expire, if they were reclassified as investigations into more serious crimes.

Mr. Samuelson called on Ms. Ny, the director of public prosecution, to drop the case entirely. “She has run this pre-investigation in such a bad manner that she must drop the whole case,” he said, criticizing Ms. Ny’s refusal, until March, to consider questioning Mr. Assange in London.

“You can’t blame Ecuador,” he said, adding that responsibility lay with “the prosecutor for having done nothing for five years.”

Kristinn Hrafnsson, a spokesman for WikiLeaks, also called for all inquiries against Mr. Assange to be dropped. “It is simply obvious that there has been no real willingness to move this forward by the Swedish prosecutor,” he said.

But Claes Borgstrom, a lawyer representing one of the complainants in the case, said he could see no reason to drop the rape investigation, adding that it was up to Mr. Assange to cooperate with the inquiry.

“Mr. Assange has now been in London for many years, probably for longer than he would have been in Sweden had he come here — even if he had been charged and found guilty,” Mr. Borgstrom said.