By the time he visits the White House next month, French President François Hollande insists he will have figured which of two women he will continue sleeping with — and call his first lady.

The embarrassed and embattled president vowed on Tuesday to make some sense of the national soap opera that has dominated his country for days:

Is his live-in girlfriend of many years, Valerie Trierweiler, still the first lady of France?

Or has he officially ditched her for his new — and, until last Friday — secret love, Julie Gayet, a gorgeous blond actress 18 years his junior, and seven years younger than Trierweiler?

The triangle took on added dimension Tuesday when a source close to Hollande claimed that Gayet is four months pregnant.

Trierweiler had been invited to accompany Hollande for a state dinner hosted by President Obama — including an overnight stay at the White House Feb. 11.

But Trierweiler, 48, has been hospitalized — with what is described as a serious case of “the blues” — since a French magazine revealed that Hollande, 59, had been sneaking out of his presidential place amid a yearlong affair with Gayet, 41.

He had been expected to come clean about his tangled affair de couer on Tuesday when he went before more than 500 reporters at his annual new year press conference.

After droning on about the dismal French economy and its record-high unemployment in his opening 30-minute speech, Hollande was asked if Trierweiler was still the first lady and No. 1 in his heart.

His answer was less than revealing.

“I understand your question and I’m sure you will understand my response,” he began.

“Everyone in their personal lives can go through tough times. That is the case [for me]. These are painful moments.

“But I have one principle: these private affairs are dealt with in private. This is neither the time nor the place to do it, so I will not be responding to any questions about my private life.”

However, he said that by the time he sits down at the state dinner with the Obamas, he will have come up with an answer to the question. Or, as he put it:

“If I do not go into detail about this today, then I will do so before the meeting which you refer to.”

Then he added that France doesn’t really have a first lady because there is “no official status” conferred on a president’s wife or partner. And, in words that couldn’t have made Trierweiler happy, he said the government money spent on a first lady should be made public — and “as small as possible.”

Trierweiler, a former journalist, has made it clear that she is eager to make the Washington trip and wants to know whether she is still Hollande’s lover, officially or unofficially.

She had lived with Hollande for years and moved into the Elysée Palace, where she has her own office and staff, after he was elected in 2012.

The White House said on Tuesday that Hollande’s trip is still on — but said questions about who would be accompanying him should be asked in Paris.

“We look forward to hosting the president of France here in February,” Obama spokesman Jay Carney said. “This is our longest and most enduring alliance.”

In a press conference that lasted more than 2¹/₂ hours, Hollande was only asked four questions about Trierweiler by a mass of reporters who seemed terrified of raising “l’affaire Gayet.”

Asked about Trierweiler’s health, Hollande said, “She’s resting and I have no further comment to make.”

For the rest of the press conference, Hollande droned on about French taxes, renewable energy, relations with Germany and other less-than-hot topics.

As for Gayet, she hasn’t spoken or surfaced since the scandal broke. But a post on the official Twitter account of the lereel.fr Web site said, “Julie Gayet is four months pregnant according to a source close to Francois Hollande at the Elysée.”

It added that this had been confirmed by a journalist at France’s M6 television news channel.

Hollande was already the least popular French president in modern history even before the magazine Closer caught him sneaking off from the palace to see Gayet. It published embarrassing photos including one of Hollande apparently disguised in a motorcycle helmet to avoid being recognized.

He told reporters Tuesday his “indignation is total” over the exposé and left open the possibility that he would sue Closer. But he didn’t deny anything the magazine said.

This was not supposed to happen to Hollande, who was elected two years ago by a French public tired of reading the latest disclosures about his flamboyant predecessor, Nicolas Sarkozy.

Hollande, who has four children from a previous relationship with a leading politician, ran as “Monsieur Normal,” the candidate who would never be associated with scandal.

Dominique Moisi, a French political analyst, said Hollande has only himself to blame.

“He wanted to impress the French with the fact that he was a normal man, that he was a man of dignity, simplicity, moral rigor,” he said. “Suddenly the French are discovering that he is like others, but in a less glorious manner, even a ridiculous manner.”