The Welsh language broadcaster wants to move to a new site, known as Yr Egin.
But the developer, the University of Wales Trinity St David, is asking the Welsh Government to give up to Â£6m.
However, the broadcaster insisted that taxpayers' cash would not be used to fund the move.
Carwyn Jones told AMs he remained supportive of the relocation idea, and a decision on whether to provide public funding would be made "soon".
He was asked about the project during a hearing of the Committee for the Scrutiny of the First Minister, held in Carmarthen on Friday.
Mr Jones said: "I remember when this was discussed with me, the reason Carmarthen was chosen was because there would be no cost to the public purse.
"Now, of course, the situation has changed.
"I have to say I made the case for the Amman Valley, because I thought it was important to go to an area where the language was under severe pressure.
"But the answer was 'Carmarthen is there, the site is there, it's not going to cost anything'.
"It's changed."
In October, Economy Secretary Ken Skates told AMs it was  "disappointing" a funding gap had appeared, but the university denied this was the case.
But a briefing paper sent by the university to the Welsh Government a month earlier stated public investment was needed for the project to "eliminate the financial viability gap".
Some campaigners have called for the move to be scrapped, and for S4C to move instead to Caernarfon.
Responding to the comments by the first minister, S4C said it was wrong to confuse the issues of relocating to Carmarthen, and the costs of the new building.
"S4C has never asked the Welsh Government for finance to fund the move - it will be cost neutral to S4C over a 20 year period," said a spokesperson for the broadcaster.
The university also stressed that its request for funding was not for the S4C move - but rather was an application "for creating infrastructure to support companies, economic and new cultural opportunities".
S4C promised there would be no cost to the taxpayer as a result of its planned move from Cardiff to Carmarthen, the first minister has said.