The CVC deal is due to receive final approval on Dec. 10 and was approved by 38 of the 42 clubs comprising LaLiga in a first round.
The alternative proposal from Spain's two largest clubs together with smaller Athletic Bilbao would see JPMorgan, Bank of America and HSBC jointly lend 2 billion euros in exchange for an fixed annual payment of 115 million euros for 25 years, a document seen by Reuters showed, confirming a report by the Financial Times.
The CVC deal would see the fund receive 11% of the league's television rights over the next 50 years in exchange for a one-off payment of 2.7 billion euros.
The vast majority of the cash is to be given to clubs to spend on new infrastructure and modernization projects as well as increasing how much they can spend on players.
Real Madrid, Barcelona and Athletic Bilbao opted out, saying the terms were too generous to CVC. Second-division Real Oviedo also rejected the deal.
In response to their alternative proposal, LaLiga's management said the CVC deal was not just about financing the league but also aimed at making it more competitive and improving its businesses.
"The CVC deal is an operation of investment in a industrial partner, not a bank credit, and (Real Madrid president) Florentino Perez knows that," LaLiga president Javier Tebas wrote on social media.
"Now he 'remembers' the clubs from LaLiga (the ones from the less interesting games), the teams he was ignoring when he was building the Super League.
"Turns out that it is Anas Laghrari (CEO of Key Capital and of the Super League disaster) who Florentino Perez sends to make a proposal with such ignorance of the legal and financial situation of LaLiga clubs that it is unfeasible. They seek confusion and continue to manage from the pub's counter."
($1 = 0.8863 euros)
(Reporting by Inti Landauro, Belén Carreño and Fernando Kallás; editing by Jason Neely and Rohith Nair)