The review - by the Committee of Advertising Practice (CAP) and the Broadcast Committee on Advertising Practice (BCAP) - has also proposed tighter controls over environmental claims and stricter rules to protect children.
The two committees will also oversee a public consultation, which closes on 19 June.
Currently, Channel 4 is the only channel where condoms can be advertised from 1900.
BCAP said the Independent Advisory Group on Sexual Health had called for a relaxation of the rules on condom advertising, after a rise in teenage pregnancy and sexually transmitted infections.
It pointed to figures which showed over 11,000 under-16s were diagnosed with chlamydia, herpes, gonorrhoea, syphilis or genital warts from 2002 to 2006.
But under its proposals, condom commercials would not be allowed around programmes aimed at children under the age of 10.
Pregnancy advice services, which includes information on abortions, would be allowed to advertise on radio and TV for the first time. Adverts for such services would have to make it clear whether it referred women for abortions.
The consultation is UK-wide, but any changes would not necessarily be applied across the UK. Broadcasters would have to tailor any advertising to the area in which it was broadcast.
TV ads for condoms outside the UK are often light hearted
Simon Blake, chief executive of Brook, which offers sexual health advice and services to young people, told BBC Radio 4's Today programme the move would help people obtain accurate information.
"Young people tell us TV is an important route through which they get a huge amount of messages about sex.
"So clear, honest, factual advertising about services which provide honest messages is clearly going to be part of shifting the balance away from this over-sexualised media."
But John Smeaton, national director of the Society for the Protection of Unborn Children, said the move would "further commercialise the killing of unborn children".
He said: "Agencies with a financial interest in promoting abortion will be in a position to buy expensive broadcast advertising, whereas groups which provide objective information about abortion and its impact on women's health will be unlikely to afford to advertise."
Liberal Democrat MP Dr Evan Harris, who sat on the science and technology select committee that examined the issue in 2007, said there was no reason why carefully regulated advice should not be available on television.
"What is vital, as the select committee recommended, is that no woman is misled by anti-abortion campaigners claiming to offer balanced advice when they never refer for abortion," he said.
"The idea that pregnancy advice, family planning and condom adverts should not appear before the 9pm watershed is as ridiculous as the former ban on tampon adverts, which have been shown not to offend despite the outcry at the time they were allowed."
The review also proposes greater protection for children from other commercials, including those for adult-rated computer games.
There would also be tighter controls to stop advertisers exaggerating environmental claims.
What stops teen pregnancies: condoms or family meals? I suspect that the problem is more down to lifestyle than to lack of information.
Advertisers could not claim their product was free of environmentally-damaging ingredient if that ingredient was not normally found in competing products, or highlight environmental benefits which were a result of legal obligations.
They would also have to base environmental claims on their product's impact across its total life cycle.
The word "free" would also have to be clarified.
Andrew Brown, chairman of CAP and BCAP, said those drawing up the code aimed to set a high bar for social responsibility.
"Our priority is to ensure that the rules remain relevant for the future so that consumers can continue to enjoy and trust the ads they see," he said.
"Throughout this process, we sought the views of industry and policymakers and now we want to hear from all other interested parties, including the people that matter the most in advertising, the general public."
The advertising codes were reviewed simultaneously for the first time.
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