Gibbons' defence lawyer said he thought it was possible she would receive a pardon from Bashir.
"An appeal from a fellow Muslim could have an impact," a government source said.
Her chief defence lawyer, Kamal al-Jazouli, said she was in a well-guarded location, separate from other prisoners.
"I have no inside information," he said. "But I think there will be a pardon. The president has the power to change sentences and pardon defendants."
He said Gibbons, who began her seventh day in detention on Saturday, was in a good room and that he had given her only a brief description of the protests to keep her from worrying.
Al-Jazouli said he believed the peers' bid to obtain Gibbons' early release had a good chance of success.
"I think the Sudanese government must now be as concerned for her safety as the British government. There are so many factions and sects and the government cannot control them. It would be a nightmare for Sudan if anyone harmed her."
Britain's Foreign Office said ministers were doing all they could to try to resolve the issue as quickly as possible.
Foreign Secretary David Miliband had spoken again to his Sudanese counterpart and voiced "our very strong concern at Mrs. Gibbons' continued detention", a spokesman said.
Gibbons let her 7-year-old pupils at Khartoum's private Unity High School pick their favourite name for a teddy bear as part of a project on animals in September. Twenty out of 23 of them chose Mohammad -- a popular boy's name in Sudan, as well as the name of Islam's Prophet.
Gibbons circulated a letter to parents, telling them that the children would be bringing the teddy bear home at weekends as part of the exercise. Two months later, a member of the school staff handed the letter to Sudan's Ministry of Education.
Gibbons was arrested last Sunday and charged on Wednesday with insulting religion, inciting hatred and showing contempt for religious beliefs -- charges punishable by up to 40 lashes, a year in prison or a fine.
She was convicted on Thursday of insulting religion.
