Montagu Private Equity is said to have entered a "period of exclusivity" in talks to buy the not-for-profit institution, with the price tag being discussed for all its assets - including property and degree-awarding powers - thought to be around £135 million.
Montagu - a newcomer to education whose current investments include Biffa waste management - is said to have outbid Pearson, the FTSE 100 education company keen to secure degree-awarding powers.
Like many traditional universities, the College of Law is a charity with a Royal Charter, and there has been some debate about whether the Charity Commission could block any sale.
However, a commission spokeswoman said it would not have to "formally approve or give legal permission for the sale of the charity's business" although it confirmed that funds from any sale "would need to be applied for charitable purposes".
She added that the commission's "permission may be required if the trustees decided to sell the charity's property", depending on "the precise terms on which the charity owned the property".
Those close to the deal reject the view that to complete any sale, the College of Law would have to give up its Royal Charter, requiring an act of Parliament.
Under the model being discussed, the College of Law would set up a company to control its assets and degree-awarding powers, then sell all the shares in that company to the private buyer.
The charity trustees would then establish a fund with the proceeds, retaining the charitable status and the Royal Charter. This fund would support the study of law through bursaries and scholarships for students at a range of institutions, thus meeting charitable objectives.
Matt Robb, senior principal at consulting firm The Parthenon Group, said there were "lots of people following this very closely in the private sector", adding that "private investors would love to invest in a traditional university".
Mr Robb said the new for-profit-owned College of Law could even decide to "rebrand itself completely" and create a separate body specialising in professional education under the same degree-awarding powers. In that event, the college would see London universities active in professional education as "targets" for competition, he added.
Meanwhile, in a separate development, the private education firm INTO is launching a £100 million worldwide investment fund for universities aimed at bridging the gap left by falling public funding. The fund could possibly support widening participation in the UK.
INTO's fund is part of the company's drive to reduce its reliance on pre-degree pathway courses for overseas students, a market hit by the government's clampdown on visas.
Andrew Colin, INTO's founder and owner, said the fund was aimed at four areas, including helping universities to establish overseas campuses and aiding widening participation by allowing the firm's facilities currently used by overseas students to be used by UK students to prepare for higher education.
The firm's approach appears to have enlisted the support of the former Labour education secretary Charles Clarke - who in a foreword to INTO's plans says "expansion of educational opportunity across the world will only take place sufficiently quickly if private and public resources are brought together in partnership".