MP Medard Segona said “substantial amendments” had been made to the bill but said he was not allowed to reveal further details.
Speaker of parliament Rebecca Kadaga recently said the bill would be passed as a “Christmas gift” to its advocates.
Homosexual acts are illegal in Uganda – this bill increases the penalties.
Foreign donors have threatened to cut aid if gay rights are not respected.
The bill, tabled by MP David Bahati, proposes longer jail terms for homosexual acts, including a life sentence in certain circumstances.
In its original form, those convicted of “aggravated homosexuality” – defined as when one of the participants is a minor, HIV-positive, disabled or a “serial offender” – faced the death penalty.
Such offences would now be punished with life imprisonment, it is understood.
The original bill also prohibited the “promotion” of gay rights and called for the punishment of anyone who “funds or sponsors homosexuality” or “abets homosexuality”.
Mr Bahati has previously said that the death penalty provision would be dropped but this has not been confirmed until now.
Mr Segona, who is on the Legal and Parliamentary committee of Uganda‘s parliament, told the BBC: “I can confirm it has been dropped.”
“Some of us who are human rights activists would discourage the death penalty,” he said.
The next step is for parliament to debate the bill. Mr Segona said he thought this would happen before the Christmas break but said he could not be more precise.
The BBC’s Ignatius Bahizi in the capital, Kampala, says that the bill has broad support in parliament.
But he notes that President Yoweri Museveni would have to sign it before it comes law – and the president may come under intense international pressure not to do so.
The bill was strongly condemned last year by Western leaders, including US President Barack Obama who described it as “odious”.
International donors have threatened to cut off aid to Uganda if the country does not do more to protect the rights of gay people.
Last month, Uganda’s speaker of parliament was involved in a row with Canada’s Foreign Minister John Baird over gay rights at a meeting of the Inter-Parliamentary Union in Quebec.
She received a rapturous welcome from several hundred anti-gay activists, including religious leaders, at Uganda’s Entebbe airport when she returned from her trip.
Correspondents say many Ugandans are deeply conservative, and say homosexuality is against their religious and cultural beliefs.