The federal government runs the online exchange for three dozen states that were unwilling or unable to do so. Many of these states have lamented the fact that they ceded control of their exchanges — and a large part of their health insurance markets — to the federal government. At the same time, some states, having seen the troubles of the federal exchange, are wary of taking on the responsibility.

Under the arrangement envisioned by Mr. Counihan, a state could set policy for its exchange while buying information technology and other services from Connecticut. Services could include determinations of eligibility, assistance with enrollment and marketing, the operation of a telephone call center and the collection of premiums from small businesses.

“This would be an ideal solution for a state that does not like big government or does not want to hire a big new staff,” Mr. Counihan said. “We have 59 employees at our exchange in Connecticut. Some states would rather have a skeleton crew of four or six or 12 employees. You can get the benefits of a state-based marketplace without the headaches of building or staffing it yourself.”

Gov. John Kitzhaber of Oregon, which is still struggling with its own insurance website, said in an interview: “We will be looking to see if there are modules we could take from another state. We are definitely looking at that.”

Likewise, Paul Hencoski, a leader of the health and human services practice at KPMG, the accounting and consulting firm, said it would make sense for states relying on the federal marketplace to “reuse the assets” developed by Connecticut, rather than starting from scratch to build their own. “Consumers would get a better functioning exchange, and states could achieve economies of scale,” he said.

When states run into problems, Mr. Counihan said, they often blame companies that supplied their information technology. But in many cases, he said, state officials are partly responsible because they did not properly manage or supervise the vendors.

Exchanges perform a wide range of functions: evaluating and certifying health plans; disseminating information about their costs and benefits; sending computer files to insurers, state Medicaid agencies and the federal government; collecting huge amounts of data needed to enroll people in health plans and calculate the federal subsidies for which they may be eligible.