As Illinois lawmakers huddle behind closed doors this week trying to find ways to plug the state’s $13 billion deficit, municipal-bond investors are sizing up how to wager amid the state’s woes.

The state’s bonds have the highest spreads — a measure of the perceived risk of default — of any state, according to Thomson Reuters data. Meanwhile, the cost of insuring against the bonds’ default keeps rising. Many muni-bond investors are avoiding Illinois even as they buy bonds of other cities and states. Others believe the state may prove a good bet, either because it will get its financial act together or be aided by the federal government.

Illinois is striking fear in bond buyers’ hearts, they say, because, more than many other financially troubled governments, the state has increasingly relied on debt to pay bills, rather than making deep spending cuts or raising income taxes to increase revenue.

On Thursday, Moody’s Investors Service said the country may be entering a period of “decreased investor appetite for municipal debt.” States most vulnerable include those that have heavily relied on debt to fund operations, such as Illinois, said the credit-rating company.

“It is a dangerous game,” said George Rusnak, national director of fixed income at Wells Fargo’s private bank, which holds about $14 billion in muni investments and isn’t an active buyer of Illinois bonds. “You can float money and hope the economy recovers. But at some point, investors say ‘enough’” and start charging the state prohibitively expensive rates to borrow more money.

State legislatures around the country are convening this month to face big budget deficits and tough decisions on taxes and spending.

Illinois, the fifth-largest issuer in the nation’s $2.9 trillion bond market, is being closely watched as a test case for how statehouses deal with these issues. The question, investors say, is whether the state will make long-term fixes that reduce risk to their investments, or simply paper over problems.

The risk premium priced into Illinois’s 10-year bonds widened in recent weeks to 2.1 percentage points above a broader muni-market benchmark. A year ago, that spread was less than one percentage point.

Bill Gross, a founder of Pacific Investment Management Co., said in an interview last week on CNBC that he is buying muni bonds but is avoiding Illinois debt.

Talks between Illinois Gov. Pat Quinn, a Democrat, and lawmakers in Springfield could extend into early next week, people familiar with the matter said. “Gov. Quinn continues to have productive meetings with legislators in an effort to stabilize the budget,” said Kelly Kraft, a spokeswoman for the governor’s budget office. “Nothing is final.”

Proposals include a $3.7 billion bond to fund the pension system and possibly an even larger “debt restructuring bond” that would allow the state to pay back vendors who haven’t been paid in months.

While current annual debt costs take up a relatively small 4% of the state’s total revenue, investors say the state’s borrowing habits indicate there is little political will to overhaul state finances, despite perennial promises to do so. They point to state pension problems as an example. Some experts have forecast that by 2023, the state could be paying out more to its retirees than it has money in the pension fund to cover those payments. That is spooking investors about buying longer-term Illinois bonds that would come due as the pension fund runs dry.

Lawmakers also are talking about raising the state income-tax rate to as much as 5% from 3% now. Investors say that move would raise about $6 billion in additional revenue, and could go a long way toward plugging the deficit. But with Republicans gaining new seats in the legislature when the new legislative session convenes Jan. 12, it could become more difficult to pass the tax increase.

Ms. Kraft said the governor is committed to funding the pensions and to enacting further pension-policy changes to reduce costs. She said the state’s additional borrowing proposal under consideration to pay vendors will “allow those owed money to preserve and create new jobs.”

Some investors say they would become buyers of Illinois bonds if the problems worsen to the point where spreads, the gap between what Illinois pays to borrow versus healthier states, widen drastically. “We become buyers when there’s a perception that it’s all going to hell in a handbasket, said Matt Dalton, CEO of Belle Haven Investments, a White Plains, N.Y., investment firm with $600 million in muni investments under management. Mr. Dalton said he has been working since July to ”eradicate“ his Illinois holdings.

One reason for the possible bottom-fishing is investors’ expectation that the federal government wouldn’t let the state default, despite a lack of precedent for that scenario. ”It would be difficult for the federal government to say yes to AIG and no to Illinois,“ said Hugh McGuirk, who oversees $16 billion in muni bonds at mutual-fund giant T. Rowe Price, referring to the $182 billion bailout of insurer American International Group. Mr. McGuirk’s funds don’t hold Illinois general-obligation bonds.

John Miller, chief investment officer of Nuveen Asset Management, has been increasing Nuveen’s Illinois holdings as part of its national muni portfolio of nearly $80 billion, because he says default fears are overblown.

He says if the governor and lawmakers can agree this week on raising the state income tax for the first time in years, it would go a long way toward closing the $13 billion deficit. In that case, he said, spreads could tighten ”quickly.“