It comes after a number of signals from the government that it is considering implementing a watershed on food and alcohol advertising.
The bill is the second to call for a TV watershed, but the first to extend proposals to non-broadcast media.
The first bill was proposed by Labour peer Baroness Thornton earlier this year. It passed two readings in the House of Lords, but it did not get any further due to lack of sufficient debating time.
Griffiths' bill will receive its first reading tomorrow. He claims to have the backing of around 50 groups including the Consumer Association, British Medical Association and the British Heart Foundation.
A pre-9pm watershed for junk food advertising on TV was one of the options considered by Ofcom in a long consultation that ran from late last year until early this year. The regulator calculated such a ban would take £211m out of broadcasters' revenues.
Instead it introduced a two-stage programme of TV restrictions, which it calculated would result in a £23m hit to broadcasters.
Ads around programmes aimed at children aged four to nine were banned from April 1 this year. This will be extended to shows aimed at children up to 15 from January 1 2008.