General Motors Co. on Friday starts taking applications from victims of accidents connected to defective ignition switches in its small cars, promising the first payouts from the about $400 million fund by November, reported The Wall Street Journal.

The nation’s largest auto maker has said it would pay from $20,000 to several million dollars per claim to cover death and injuries from crashes in certain Chevrolet Cobalt, Saturn Ion and Pontiac vehicles in which the air bags didn’t deploy. A death would automatically be awarded $1 million for pain and suffering above any other payments.

The company has been criticized for excluding other midsize and larger GM cars that also face recalls for faulty ignition switches that can turn off air bags and impede brake and power steering. In July, the company recalled 8.5 million Chevrolet Impala and larger vehicles that it said also had deaths due to air bags not activating in crashes. GM has said the circumstances are different and it has no evidence the defect led to the crashes.

Attorney Bob Hilliard, who plans to submit claims for some clients, insists GM hasn’t gone far enough with its plan. On Wednesday, Mr. Hilliard filed a lawsuit on behalf of 658 people he claims are ineligible to apply due to the auto maker’s rules.

“To me the whole thing is a shallow and superficial response and designed to save GM money,” Mr. Hilliard said.

About a dozen claims are expected to come during the first week, said GM compensation adviser Kenneth Feinberg. He said the program doesn’t have a financial cap. Most claims will be filed by attorneys whose clients want to gauge what the company’s out-of-court settlement would provide.

“It is a free preview, so why wouldn’t they do it?” Mr. Feinberg said in an interview. “If they are satisfied with the amount, they yield the right to sue and we send them a check. If they aren’t they can go and litigate.”

One victim who expects to file a claim is Caitlin Pereira, 28, who was involved in a car crash in her 2005 Saturn Ion in September 2013. Ms. Pereira was driving on a Los Angeles freeway when she collided with a dump truck. The air bag didn’t deploy and she suffered a broken nose and a concussion that she said has affected her memory.

Ms. Pereira said she has spent about $15,000 out of pocket on medical bills, which has been a hardship. “I ended up missing out on four months of work and had to relocate back home in Connecticut. I am now back to work at a new job, but the accident has set us back financially,” she said of her family. “I am just trying to recoup what I can so we can get back on our feet.”

In the 2.6 million compact cars covered by the plan, GM has said jarring the ignition key could move the switches to accessory from run, cutting power to the engine, shutting down electric steering and stopping the air bags from deploying.

Mr. Feinberg, who has sole authority over claim approvals and payouts, said he has no idea how many claims will be filed and processed by Dec. 31, when GM stops taking claims.

“If history is any indication, there are two spikes, with the first coming in the first two or three weeks and then a flood of claims before the program closes at the end of the year,” Mr. Feinberg said.