"The two work in a beauty parlour and developed affinity during the last four years. While one of them belongs to Mant area, the other is from Siddharth Nagar," a police officer said.
"On Sunday evening, both the girls contacted Harendra Pal Singht, SHO Jamuna Paar police station, and sought his help for their marriage. They said they were facing opposition from family members," Deputy Superintendent of Police Rakesh Kumar told reporters.
"Both the girls were counselled to change their decision by police as well as by their family members, but, at present, they appear adamant," he said.
Mr Kumar said the girls have gone back to their families for now.
Same sex marriages are illegal in India and homosexuality is a crime under Section Section 377 (unnatural offences) of IPC.