Ministers have already been forced to rewrite the bill following objections by peers to the original wording.
It had stated that children born through fertility treatment needed a "social network", not a father and mother.
The Lords will vote on the revised bill on Tuesday.
Those pushing for the role of fathers to be recognised as vital include Baroness Ruth Deech, a former head of the Government's fertility watchdog, the Human Fertilisation and Embryology Authority (HFEA).
However fertility treatment pioneer Lord Winston said that the call for a such a "father clause" would discriminate against career women.
He said: "This is prejudiced against women who have paid their taxes, been well educated and delayed motherhood because they are working hard.
"If they are left without a partner and still want a baby [with a donor], this would prejudice their treatment and ask doctors to make a value judgment about them."
Lawyer Natalie Gamble, an expert on fertility regulations, said: "Equality rules mean clinics have to give treatment to single women and same sex couples anyway. The Government doesn't need to enshrine this in new laws."
Annulments are usually heard at county court, but can go to the High Court if cases are particularly complex.
Marriages can be annulled if one of the parties was under 16 at the time of the wedding, if it was a bigamous union, or if the couple are closely related.
The latest figures, for 2005, show 251 marriages were annulled that year.
There is no breakdown of how many cases fell into each category.