PHILADELPHIA -- For weeks, the Phillies have said they plan to make modest moves to improve their roster before Spring Training.

But, they added, they will keep their options open.

It explains why they have pursued free-agent right-hander Yoshinobu Yamamoto this offseason, even after they signed Aaron Nola to a seven-year, $172 million contract last month. The Athletic reported on Wednesday that the Phillies made Yamamoto an official offer, as expected. They met with him last week in Los Angeles. They felt good about the meeting. They are contenders to sign him, even if they are not considered favorites.

The Yankees, Mets and Dodgers are widely viewed as the favorites, with the Giants, Red Sox and Blue Jays also interested.

Sources have said since last month that Nola’s deal never precluded the Phillies from signing Yamamoto, even if they characterized those chances as unlikely, saying the Phils would need to clear payroll to make Yamamoto fit into their budget. But the Phillies have been genuine in their interest, getting Bryce Harper involved a couple times to make a sales pitch for Philadelphia, including a FaceTime chat with Yamamoto last week.

Asked last month after Nola’s press conference about what could come next for the Phils, Phillies managing partner John Middleton said, “Dave [Dombrowski] comes to me and says, ‘This is my opportunity. This is what I think of that opportunity. This is what the opportunity costs.’ Then he says yes or no. Basically, everything that Dave has recommended to me, we’ve done. But I’ll also go to him and say, ‘Dave, would you want more money to do whatever?’ Sometimes he’ll say yes. Sometimes he’ll say no. Sometimes he’ll say, ‘I’ve got to think about it.’

“There’s a budget. There’s more money to do stuff. And budgets can be changed.”

It depends on the player?

“It’s the opportunity,” Middleton said. “How good is it?”

Yamamoto is a really good pitcher, but how far are the Phillies willing to go? And will it be enough to beat the Yankees, Mets, Dodgers and the rest?