Manuel Pellegrini has suggested Frank Lampard could extend his stay at Manchester City beyond this season after the player’s precise status with the Premier League champions was thrown into confusion and he came under fire from supporters of New York City “outraged” by his decision not to join the Major League Soccer club, as agreed, in the new year.

Lampard, who scored a 73rd-minute winner in City’s 3-2 victory over Sunderland on Thursday, had initially been set to join NYC after 31 December when it was believed his loan contract at the Etihad Stadium expired. However, it has emerged that Lampard has never been on loan at City from NYC but instead signed a deal that extended beyond New Year’s Eve. The champions had to have the existing terms amended to have a break clause removed and were granted permission to do so by the Premier League on Wednesday.

As no player can be registered to two clubs at the same time Lampard is officially on City’s books. He may possibly have signed a pre-contract agreement or agreed terms with City Football Group, the parent company of both clubs.

After the win against Sunderland, Pellegrini was forced to defend City from angry criticism directed by NYC’s independent supporters group, the Third Rail. He said: “That is why every time you ask me about Frank Lampard during December, I repeat once again we are trying to keep Frank but it is not easy because others are involved in this situation.

“Maybe all the New York fans, and I understand perfectly, want to see Frank playing because for the fans in New York to see Frank play is an amazing thing. I hope that he will be there next year but I think that in this moment the best decision for him, for the club, for New York is to stay here with us.”

When pressed why he had said only “hope” regarding Lampard playing at NYC next season, Pellegrini refused to clarify that the midfielder would definitely do so. Pellegrini said: “You are talking May or June? You cannot have all different questions about Frank Lampard. He will stay now with us until the end of the season. In May maybe you can start thinking about what will happen in the future. He will stay with us until May.”

Asked if there was any chance Lampard could move to NYC before the end of the season, Pellegrini added: “No. When you asked me before we qualified for the Champions League if there is some link between the situations, I always said no and Frank knows exactly what I think about that. He will stay here until the end of the season, no problems about that.”

The statement from the Third Rail said: “Many fans, including our members, decided to support the team, committed to season tickets and bought merchandise under the impression that Frank Lampard would be playing for New York City Football Club, not Manchester City. Many of those fans are rightly outraged by this decision.”

The Sunderland manager, Gus Poyet, had no issue with Lampard being available to play for City against his team. “For me it’s clever, it’s smart. It’s legal, so I’ve got no problem with that,” he said.

For the second match in a row City ceded a two-goal lead, after strikes from Yaya Touré and Stevan Jovetic were cancelled out by Jack Rodwell and Adam Johnson later in the second half before Lampard’s winner.

Despite Pellegrini having only three frontline strikers in Jovetic, Sergio Agüero and Edin Dzeko, and the latter two currently being injured, he played down whether a fourth will be recruited in the transfer window. “I think we are not thinking about the window,” he said. “We have the option. We will see players we can bring or sell or go on loan but it is not our first important thing at this moment. We will see during January what will happen. So we have an option and we can use it until the end of the transfer window.”

City have been linked with a move for Swansea City’s top scorer Wilfried Bony. “Every week I read and hear so many players come here to Manchester City,” said Pellegrini when asked about the forward. “I once again remind you that we have important restrictions about the amount of money and restrictions on players we can put in our Champions League squad [due to FFP sanctions]. So it’s difficult to bring in a player when he cannot play in Champions League, which is why I repeat we will see during January which is the best for our team.”