Christie campaigned for Baker in 2010, when Baker lost to Gov. Deval Patrick, the incumbent Democrat. Baker is facing a pesky challenge from the right this time, while his likely Democratic opponent, state Attorney General Martha Coakley, has drawn attention to the New Jersey donation.
The controversy brings fresh scrutiny to New Jersey's pension system, which became the center of debate this month when Gov. Christie helped close a $1 billion revenue shortfall for the fiscal year ending June 30 by cutting the state's payment into the pension fund to $696 million, down from $1.58 billion.
New Jersey has committed about $380 million of the pension fund to a handful of venture-capital firms, which typically invest in start-up companies.
Christie is also chairman of the Republican Governors Association, which raises money to help elect Republican gubernatorial candidates.
At issue in Treasury's review is whether Baker is an "investment management professional" for General Catalyst, Christopher McDonough, director of Treasury's Division of Investment, wrote Wednesday in a memo to the investment council.
The council's policy prohibits Treasury from investing in a fund whose investment managers have donated to New Jersey political parties or committees in the preceding two years.
The review was prompted by reports by the website PandoDaily.
Coakley has called for a Securities and Exchange Commission review.
"Charlie's contribution was completely transparent and permissible, and any objective review of the facts will demonstrate that fact," Baker campaign spokesman Tim Buckley said in an e-mail. Baker also commissioned an independent review by Anthony Herman, a lawyer with Covington & Burling L.L.P. and former general counsel to the Federal Election Commission.
Herman's opinion, posted online by the Boston Globe, concluded Baker's "contribution to the New Jersey Republican State Committee violated neither the spirit nor the letter of the 'pay-to-play' rules."
In a statement, General Catalyst said: "To be clear, General Catalyst did not 'pay to play.' There has been no claim by anyone that Charlie Baker was involved in persuading New Jersey to invest in a General Catalyst fund or that anyone on the New Jersey pension board was aware of Charlie Baker's contribution to the New Jersey Republican State Committee or caused the pension fund to invest because of the contribution."
Baker is not involved in investment management decisions, the firm said. Rather, as an executive-in-residence since April 2011, he brings executive and operational expertise to operating businesses and mentoring executives, the firm said.
It added that the firm did not require Baker to disclose the donation because pay-to-play policies only apply to investment management personnel and those who solicit investors.
In 2013, General Catalyst said, Baker joined the boards of Mulberry Health and that firm's holding company, Oscar Insurance. General Catalyst invested in the companies.
New Jersey pension funds represent 3 percent of the General Catalyst fund, a company spokeswoman said.