The BBC understands Australian Lynton Crosby was targeted by backers of "The Know" campaign over the summer.
It is claimed CTF Partners declined the offer after a two-week period.
A spokesman for CTF Partners neither confirmed nor denied the claim, telling the BBC "we never confirm who we work for or who approaches us".
The Know campaign was set up and part-funded by UKIP donor Arron Banks. It announced on Friday that it is rebranding with the new name Leave.eu.
The BBC understands representatives from campaigns on both sides of the debate about the UK's EU future have approached the firm run by Mr Crosby, who helped devise the Conservatives' successful election campaign.
They were rebuffed because he believes it is too early to commit to work with either side ahead of the government's renegotiation of its membership.
A senior source at The Know campaign said they believed the offer was given serious consideration but declined out of loyalty to the prime minister who will campaign to stay in a reformed EU.
Another very well-placed figure in that campaign said they were not aware of specific sums ever being discussed but it may have come up over a "convivial lunch".
The Know campaign announced on Thursday that it had signed up US-based campaign strategists Goddard Gunster to advise it on the referendum.
A Conservative election strategist was approached and his firm offered £2m to work for one of the No campaigns in the upcoming EU referendum.