Defeat on Wednesday left Tottenham in 14th place in the table, piling pressure on manager Thomas Frank, and disgruntled fans let the players know their feelings, with the likes of Micky van de Ven and Pedro Porro involved in verbal exchanges.
Argentina's World Cup-winning defender Romero apologised to the fans in a social media post and suggested it should not be the players having to front up to the supporters.
"We are responsible, there's no doubt about that. But we will keep facing up to it and trying to turn the situation around, for ourselves and for the club," he said.
"At times like this, it should be other people coming out to speak, but they don't - as has been happening for several years now. They only show up when things are going well."
Read more about Tottenham's spiralling crisis
Tottenham chairman Daniel Levy stepped down as executive chairman in September after 25 years, with chief executive Vinai Venkatesham taking on the day-to-day running of the club.
It was not clear who Romero was referring to in his Instagram post in which he apologised to the fans for another defeat in what is turning into a disappointing season.
But the 27-year-old insisted the players will stick together and try to emerge from the current slump, which has seen them win only twice in their last 12 Premier League games.
"We'll stay here, working, sticking together and giving our all to turn things around," he said.
"Especially at times like this, keeping quiet, working harder and moving forward all together, is part of football."
Frank says Romero 'made a mistake'
Tottenham manager Thomas Frank said Romero was a "young leader" who made a mistake, adding that it was better to deal with the club's issues internally.
The club's sporting director, Johan Lange, and Frank had met with Romero to discuss the matter, the manager told reporters on Thursday.
"When you're a young leader, sometimes you make a mistake. It's, of course, good to keep it internally. Johan and I had a good conversation with him this morning about everything. We keep it internally," he said.
Tottenham host Aston Villa in the FA Cup third round on Saturday.