The Bee Gees singer, who passed away last week aged 62, had a baby with former housekeeper Claire Yang in November 2008.
Robin's wife Dwina had apparently given her blessing for the relationship at the time, but asked Yang to move out when she fell pregnant.
She is now worried that Yang will demand more of Robin's estimated £140m fortune and has called lawyers in. The recent £4m payout came after a year-long legal battle, the Sunday Mirror reports.
"If Claire does go to the funeral she will be keeping her distance from Dwina and it will be awkward. Robin never walked away from his responsibilities and always made sure the child was fully provided for," a source said.
"Snow Robin is a Gibb child so will be treated exactly the same as the other three children. It is not about the money but it is the nature of the lengthy legal process which distressed Dwina and the family.
"Relations between both sides have become very strained and Dwina is aware that it is unlikely to be the end of the matter. It could get very messy and distressing."
Robin is said to have bought Yang a five-bedroom, £850,000 house 25 miles from his home two years ago.
The house, where Yang lives with her daughter, was transferred into her name on April 19, four weeks before Robin died, according to Land Registry records.
"Claire wanted to spend her life with Robin. She didn't just want to share him with Dwina. But this was never going to happen. Robin and Dwina were never going to split up because they loved each too much," the source added.
"In fact having a child with Robin made it worse for Claire because she was unable to spend as much time with Robin. But Robin provided everything for Claire and the baby, a house, car, nanny - everything. He would not let the child go without."
A private funeral will be held for Robin on Friday, June 8, followed by a public memorial service later in the summer.
Photo gallery - Robin Gibb's career in pictures: