Zlatan Ibrahimovic said he will "see what happens" about extending his stay at Manchester United - but did reveal his children pleaded with him to join the Old Trafford club last year.
Ibrahimovic, 35, left Paris St-Germain in July, signing a one-season deal with an option for a second year.
United boss Jose Mourinho believes the Swede will stay - but added: "I never beg for a player to sign a contract."
"We have another two months of the season to go," Ibrahimovic said.
"Let's see how I feel, the situation. Somebody made up a story that if we don't qualify for the Champions League I will not extend. It has nothing to do with that."
After Ibrahimovic scored the winner in Sunday's 3-2 EFL Cup final victory over Southampton, Mourinho said "the fans can go to the door of his house and stay there all night" to convince the striker to stay.
Ibrahimovic said his "special relationship" with the Portuguese, who also managed him at Inter Milan, was key to his decision to join United - but was not the only factor.
"My mind was not here, then my kids started to bump my head - even they wanted to see me play at United," he said of the months leading up to his departure from French champions PSG.
"Then Jose called. When he called, it was basically: 'Tell me what number I should wear.' My kids are satisfied with what I am doing, but this time I am the boss, not them."
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Ibrahimovic collected the 32nd trophy of his career after heading an 87th-minute winner at Wembley, having earlier given his side the lead with a brilliant 19th-minute free-kick.
Jesse Lingard put United 2-0 in front before Manolo Gabbiadini scored twice for the Saints to level, after having an 11th-minute effort contentiously ruled out for offside.
Eighteen years on from making his professional debut for Malmo, the former Ajax, Juventus, Inter Milan, Barcelona and AC Milan striker was asked whether he has thought about retirement.
"I came here when people thought it was impossible for me to do what I am able to do. It feels good. I am enjoying it," he added.
"I always want more. This is my 32nd trophy. I've been in five different countries, I've been in the best clubs in the world and I'm repeating every year what I am doing.
"I will stop on top. If I don't perform, I will not play. I will not be like others, still playing because they are who they are. I will play as long as I can bring results."
Ibrahimovic has scored 26 goals in 38 games this season and is United's top scorer. Midfielder Juan Mata is behind him in second, with nine goals.
"Zlatan won the game for us because he was outstanding," Mourinho said after Sunday's win secured his first major trophy as United boss.
"When he went to Barcelona [from Inter, in 2009], I was very sad. I know the potential. Only a silly player comes to England if he doesn't feel he can do it. Who better to know? Him. Not me or you.
"When he decided to come here it is because he feels ready. It is not my credit. It is him. Nothing for me."
Zlatan's 'house of medals'
Question from journalist: Is your fitness a natural fitness?
Ibrahimovic: "I feel fresh. I feel good. I'm an animal. I feel like a lion."
Why are you like a lion?
"I am a lion. I don't want to be a lion."
Do you mean you have the hunger of a lion?
"The lion is born a lion."
What does that mean?
"It means I'm a lion! I never talk so much with journalists. I never stopped so long even for the French people."
You say you've got 32 medals what do you do with them?
"It is in the museum. I have a house only for the medals."
'Echoes of Cantona' - analysis
BBC Sport chief football writer Phil McNulty
Mourinho's move to bring Ibrahimovic in on a free transfer from Paris St-Germain was strategic and wise. He is a personality of equal stature and confidence, had a point to prove having never played in England and could provide the sort of charisma that had echoes of the great Eric Cantona.
How United needed Ibrahimovic on Sunday because for long periods they were desperately average, outplayed by Southampton and had their hand held by Lady Luck throughout.
If United are to build on this first trophy of the Mourinho era, Ibrahimovic's continued presence is essential because the EFL Cup final win is only the first building block in an edifice that requires considerable renovation after the dismal post-Sir Alex Ferguson years of David Moyes and Van Gaal.
Zlatan's cup final record - the stats
- Ibrahimovic has scored six goals in his past five domestic cup finals (adding to four goals in four with PSG)
- He has scored more than any Premier League player this season - 26 goals in all competitions
- His opening goal was the first Southampton conceded in this year's EFL Cup - after 468 minutes of football