A woman and child, both conceived in that way, want access to non-identifying information about the donors and, ultimately, for a voluntary contact register to be established.
Monica Carss-Frisk QC, who is representing both claimants, told the High Court on Wednesday having information about their fathers would save them from the "anger, frustration and hurt" of not knowing.
The case follows calls last week from a fertility expert for children who are conceived with donor sperm to be allowed to trace their biological fathers.
The government is currently carrying out a consultation exercise to determine whether the existing laws should be changed.
Lack of information
The High Court challenge is being made by 29-year-old Joanna Rose, who lives in Australia, and a mother on behalf of her six-year-old child, neither of whom can be named for legal reasons.
Under the terms of the 1990 Human Fertilisation and Embryology Act, children who are born through donor insemination are entitled to information on the height, hair colour and race of their father when they reach 18.
However, Ms Rose is not entitled to those details because she was born before the legislation came into effect.
Ms Carss-Frisk told the Justice Scott Baker: "She has for a long time sought to obtain more information about her donor in order to attempt to resolve her feelings of incompleteness, anger, frustration and hurt."
She said Ms Rose had recently discovered the records relating to her conception had been destroyed, but wanted to continue with the case to give others more rights than she had.
Ms Carss-Frisk said that the parents of the six-year-old, who are from York, were trying to give their child all the information possible about her origins.
"They are frustrated in so doing by the lack of information available to them to answer the inevitable questions their daughter has about her donor."
She said there was no question of compulsory disclosure of information.
But she added the lack of a voluntary mechanism for exchange of information and contact breached her clients' human rights.
"There is no justification for the state's failure to take those limited steps, which do not involve the compulsory disclosure of the identity of the donor."
Striking a balance
Lawyers are quoting Article 8 of the Human Rights Act, which they argue guarantees respect for private and family life, including the right to form a personal identity.
They will also quote Article 14, an anti-discrimination provision in the Act, arguing that donor offspring should have the same rights as adopted children to trace their biological parents.
The High Court hearing is to establish whether Article 8 is applicable in this situation.
An earlier ruling stated part of the case will not be considered until after the current government consultation on issues raised in the action has been published, which could be February 2003.
The cases are being backed by the campaign group Liberty.
One of its lawyers Joanne Sawyer said: "We hope this case will open up the possibility of allowing donor-conceived adults and children to have information about their genetic make-up and background.
"The law has to strike a balance between their rights and donors' rights to privacy."
The hearing, expected to last three days, was adjourned until Thursday.