Supporters, including families of the six men, handed in the petition urging Prime Minister David Cameron to intervene.
They have also launched a campaign to raise money for a "fighting fund" to help their loved ones in prison.
Billy Irving, from Connel, Scotland; Nick Dunn, from Ashington, Northumberland; Ray Tindall from Chester; Paul Towers, from Yorkshire; John Armstrong from Wigton, Cumbria, and Nicholas Simpson, from Catterick, North Yorkshire, were arrested on October 12 and have been in prison in India since October 24.
The men were working for US private maritime company AdvanFort providing anti-piracy protection when their ship was detained.
According to the men, Indian authorities claim the vessel entered Indian waters illegally with weapons on board, despite AdvanFort apparently insisting the ship had the correct papers.
A petition, set up on online platform Change.org, calls on the Government to step in and help the men.
It was delivered to Downing Street today by family members including Mr Irving's partner Yvonne MacHugh as well as Mr Dunn's sister Lisa Dunn.
Ms Dunn said: "My brother has been in prison all this time. I went to India to see him a few weeks ago and the situation is just terrible.
"We are desperate to hear what is going to be done, and now that we have collected over 137,000 signatures in the UK we will be seeking to appeal to the Government to debate this in the House urgently."
Maritime welfare charity The Mission To Seafarers has been providing prison-visiting services to Chennai prison where the men are being held and working to support families in the UK.
According to the charity a bail hearing for the men will be held on Thursday or Friday.
The organisation is helping to manage a fundraising campaign in support of the men.
Rev Andrew Wright, secretary general, said: "We have set up a mechanism to enable families to fundraise in support of the imprisoned men.
"The Mission To Seafarers has been closely involved with both the crew and their families throughout this long ordeal and we will continue to do all that we can in support.
"Any donation made to the fund will be used in direct aid to the imprisoned crew across a range of areas, including adequate medical provision, food and water, clothing and counselling."