Updated

The Obama administration faced renewed criticism Thursday for entering nuclear talks with Iran without first demanding the release of U.S. citizens, following reports that imprisoned pastor Saeed Abedini is facing threats on his life from other prisoners.

"It is unconscionable that senior American diplomats, including the secretary of State ... could not bring themselves to even mention his name, or those of fellow Americans detained in Iran," Sen. Ted Cruz, R-Texas, said in a statement.

The State Department earlier faced criticism after a six-month deal with Iran was announced that rolled back sanctions in exchange for a pause in parts of Iran's nuclear energy program. The department acknowledged that the status of Abedini, and other American prisoners, was not part of those discussions.

On Tuesday at NATO headquarters in Belgium, Kerry defended the department's efforts, saying the release of American prisoners in such countries as Iran, North Korea and Cuba remains a high priority.

"One day is too long to be in captivity, and one day for any American citizen is more than any American wants to see somebody endure. This has been too long in every case, and we will do everything we can and continue to," he said. "But these things are often best resolved in quiet diplomacy, under the radar screen, behind the scenes, and that is exactly what we have been pursuing."

He said that "when, in fact, we secure their release, the track record of those outreaches and those initiatives will speak for themselves."

But Jay Sekulow -- with the American Center for Law and Justice, which represents the Abedini family -- wrote on FoxNews.com that "our diplomacy is worse than 'quiet.' It's inept."

The ACLJ claimed Tuesday, after Abedini's family visited him at Iran's Rajai Shahr prison, that he's facing "constant threats" on his life. The group said that since being transferred to that prison, the pastor has woken up "several nights" to see "men standing over him with knives."

According to the ACLJ, the pastor has been robbed repeatedly, has suffered from internal bleeding as a result of beatings at the last prison, and is once again being deprived of critical medication.

Further, according to the group, he is "covered" in lice.

Cruz, in his statement, noted that President Obama personally raised Abedini's case in a historic phone call with Iranian President Hassan Rouhani.

But he said: "The president's action should not have been an end unto itself but rather the first step in a sustained effort to get these men home. Their unconditional release should be central, not peripheral, to any further agreement with Iran."

Abedini, a U.S. citizen originally from Iran, has been imprisoned since late 2012. Earlier this year, he was sentenced to eight years in prison.

His case is one of several drawing the attention of not only groups like the ACLJ but also Congress. Former FBI agent Robert Levinson has been missing for six years after visiting an island off the coast of Iran, and former Marine Amir Hekmati is also in an Iranian prison, and supporters have been appealing for their release.