WASHINGTON — After a setback in the Supreme Court in the Hobby Lobby case, President Obama is facing mounting pressure from religious groups demanding to be excluded from his long-promised executive order that would bar discrimination against gay men and lesbians by companies that do government work.

The president has yet to sign the executive order, but last week a group of major faith organizations, including some of Mr. Obama’s allies, said he should consider adding an exemption for groups whose religious beliefs oppose homosexuality. In Burwell v. Hobby Lobby Stores, the court ruled that family-run corporations with religious objections could be exempted from providing employees with insurance coverage for contraception.

The demands of the faith organizations pose a dilemma for Mr. Obama, who has struggled to preserve freedom of expression among religious groups while supporting the rights of gay men and lesbians. Mr. Obama could unleash a conservative uproar if he is seen as intruding on religious beliefs, but many of his strongest supporters would be bitterly disappointed if he appeared to grant any leeway to anti-gay discrimination.

The White House has given no reason for the executive order’s delay.

In a July 1 letter to Mr. Obama sent the day after the Hobby Lobby case was decided, leaders of religious groups wrote that “we are asking that an extension of protection for one group not come at the expense of faith communities whose religious identity and beliefs motivate them to serve those in need.”