When Tom Callahan, a Vermont-born lawyer living in Moscow, agreed to be the Bachelor of the Week for a Russian gossip website in 2016, he figured it would be a harmless, if not silly, pursuit.

Dana Callahan, who was born in Kazakhstan and grew up in Russia after the dissolution of the Soviet Union, spied a behind-the-scenes photo of Mr. Callahan on Instagram and clicked “like.” The DMs flew, and before long they were an item.

“Dana never saw the article,” Mr. Callahan, 39, recalled. “I didn’t even let her see it until we’d been together for years, it was so embarrassing.”

They spent the next two years living together in Moscow. After Mr. Callahan’s contract with his law firm ended, the couple decamped to New York City, where Mr. Callahan had lived on and off since 2003, and married. Ms. Callahan, now 32, learned English and got her master’s degree in fashion history and textile studies from the Fashion Institute of Technology.

The one-bedroom, rent-stabilized apartment they shared in an Upper West Side brownstone was the stuff of New York real estate dreams, with its proximity to Central Park, soaring ceilings and decorative fireplace. After six years, though, they realized that if they wanted to start a family, lofting their full-size bed and tucking a crib underneath wouldn’t cut it.

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To buy a larger, comparable home in Manhattan, Mr. Callahan, who works at a finance-focused law firm, figured they would have to spend north of $1 million and save for at least another five years. “I’m not from a lot of money,” he said. “The common knowledge I always heard was that for below $800,000 in Manhattan, you’re not going to find anything.”

So they were surprised when they began perusing real estate sites and found listings for well below $800,000. “It was kind of a eureka moment,” Mr. Callahan said.

When Ms. Callahan found out she was pregnant last winter, the epiphany morphed into a deadline.

The couple wanted a spacious two-bedroom apartment — a place that didn’t feel like a downgrade from their rental, but at a price that wouldn’t cramp their lifestyle. To find it, they knew they would have to take their $700,000 budget beyond the Upper West Side.

“I wanted to have bigger windows,” Ms. Callahan said. “I wanted to have a good-sized living room” — things she didn’t have growing up.

She added: “But my dream was just to have a king-size bed.”

For help, they contacted Sargis Mosyan, a broker with Core Real Estate. “If you go a little bit higher above Central Park — Morningside Heights, South Harlem area — the prices change a lot,” he said. “And for the same price, you can get a lot better space than, let’s say, on the Upper West Side.”

Among their options:

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