The state's Republican Governor Asa Hutchinson has stirred intense controversy since announcing the executions of eight men in 10 days, citing shortages of a drug used in lethal injections.
The inmates said the accelerated schedule did not give them enough time to prepare their appeals, adding that the shortened timeframe violated Arkansas rules.
"Each plaintiff named in this suit has suffered irreparable injury on account of the frantic execution schedule sought by the state," six of the eight men concerned wrote in their complaint.
"Not only has Arkansas's rush to execute eight plaintiffs over the course of ten days forced the state to violate its own mandatory procedures for clemency review, but it has necessarily undermined each plaintiff's individual opportunity to advocate for mercy."
The eighth amendment of the US Constitution prohibits cruel and unusual punishment.
Death penalty critics have condemned the executions, noting that Arkansas has not conducted a single execution since 2005.
The accelerated schedule comes amid a shortage of death penalty drugs across the United States, with Arkansas's stock of midazolam, a sedative used to anesthetize inmates, nearing its expiration date.
The inmates also complained about the state's drug protocol, saying midazolam does not make recipients sufficiently unconscious, meaning they suffer extreme pain.
Under the governor's plan, two of the inmates would be put to death on April 17, two on April 20, two on April 24 and another pair on April 27.
No single US state has held eight executions in 10 days since the Supreme Court reinstated the death penalty in 1976.
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