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.
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 the city. In Dun Laoghaire, the sign-up rate was 41pc.
The figures come as public registration for the under-sixes free GP care begins today, to facilitate the commencement of the new service from July 1.
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.
Stephen McMahon of the Irish Patients' Association told the Herald that 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.
"What stress test is being done to see the impact on everybody over the age of six for those practices - in other words, the elderly, the chronically ill, the other end of the population, and that is my concern with the low uptake to date of the GPs signing on," he said.
However, Mr McMahon said that he believes it has be done in the spirit that there is not unfair pressure put on GPs to sign up.
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.
"Contracts will continue to be accepted as the scheme runs," she said.
The online registration system - www.gpvisitcard.ie - will be available from today for parents of children under six.