Romero has become Inter's leading defensive target this summer, with the club working on two fronts: talks with Tottenham over a fee, and parallel discussions with Romero's representatives, who travelled to Milan at the same time sporting director Piero Ausilio returned from a trip to Germany.
Speaking at the 23rd Magna Graecia Film Festival in Soverato, Ausilio confirmed to Sky Sport that Inter are pursuing the deal.
"There's definitely interest that we're showing the player and the club. These things take time, but there's certainly interest — we'll see," he said.
Inter Milan want to secure Romero's agreement before pushing ahead with Tottenham, hoping a deal with the player strengthens their hand in talks over his valuation.
Ausilio also gave a wider assessment of Inter's ambitions for the season. Asked about the club's targets, he said: "I think Inter have never hidden from it, we always try to do our utmost.
"And if doing our utmost means, like this year, winning two titles, that's a great thing. There's no season where Inter start out and can be satisfied with Champions League qualification or a mid-table finish. We'll try to do everything we can to bring something home — sometimes it works out, sometimes it doesn't, but we'll always give our maximum."
On the search for a successor to Denzel Dumfries at right-back, Ausilio played down any imminent arrival: "There's no name yet, we're doing our evaluations. Unfortunately Khalaili had this issue and it slowed us down a bit, but that's part of football. People forget we're still only at the end of July and there's more than a month left until the transfer window closes. We have all the calm and peace of mind we need to gradually find the right project and the right player for what we need. Perisic? Absolutely not."
Asked about the loan sale of Matteo Cocchi to Padova, which included an option and counter-option, Ausilio praised the buying club: "He was also close to joining Catanzaro, who are fortunate to have a very good ownership and sporting director. I say that because I've had the chance to get to know them — you have to be proud to have a club like that. Cocchi was an opportunity, then he went to Padova, but there will certainly be other players for us to work together on in future."
On whether Benjamin Pavard will stay at the club, he added: "We'll see if he stays, those are evaluations we'll make further down the line — we're still at the start and have all the time we need. There are areas of the squad where Inter are certainly in a stronger position than others. Up front, for example, we have four very strong attackers. So I don't think there will be changes there."
Finally, on captain Lautaro Martínez, who lost the World Cup final with Argentina earlier this month, Ausilio said: "I speak to Lautaro often. He's obviously disappointed with how the final went, but he's just as motivated to come back and immediately do well for his club. He's Inter's captain, and I can hear how motivated he is. Right now it's right that he rests a bit, which he's doing with his family, and soon he'll join up with the squad and do the great things he's always done and that he's got us used to."