Health Minister Leo Varadkar believes that up to 90pc of GPs will sign up to the scheme to provide free care for the under-sixes by the end of the year.
Public registration for the scheme began today to facilitate the commencement of the new service from July 1.
This morning, registrations were being made at a rate of 60-per-minute - causing the website to crash briefly - and 3,000 had signed up by 9am.
By the close of business on Friday, just 59pc of GPs had signed up for the contract.
Rates vary from region to region and across Dublin. In Dun Laoghaire, the sign-up rate was 41pc.
However, Mr Varadkar said he was confident that families would be able to access the scheme.
"It was less than a third two weeks ago, it'll be two thirds this week, I'm sure of that, and I think we'll have 80 or 90pc signed up before the end of the year," he said.
But this morning the Government said that it anticipated there wouldn't be complete cover for the entire country.
Minister of State, Kathleen Lynch, said the Government is working on how to provide free GP care for the under-sixes in parts of the country where very few doctors have signed up to the scheme.
Speaking on RTE's Morning Ireland, Ms Lynch, said the Government has been working on what can be done when parents cannot access a GP within a reasonable distance.
Stephen McMahon of the Irish Patients' Association said his concern is that, from a patient point of view, if only half of GPs have 100pc of the under-sixes, what will happen with the over-sixes for those practices.
A spokesperson for the HSE told the Herald that there is no cut-off point for when GPs can sign up for the under six contract scheme, which can be accessed at www.gpvisitcard.ie