The Deputy First Minister said Labour’s call for the legislation to be limited to children under 16 could allay wider public concerns about the scheme.
The Children and Young People Act enabled health providers and councils to appoint a single point of contact to monitor the wellbeing of everyone up to 18.
Opponents have challenged the Act in court, insisting the “state guardian” would breach human rights on private and family life.
The Supreme Court has forced the Scottish Government to review the law, ruling the requirement to share information on child welfare between public bodies could lead to widespread dissemination of confidential data.
The Liberal Democrats, who initially backed named person, have now described the scheme as “broken” and called for a wide-ranging independent review beyond the “narrow, technical issue” identified by the Supreme Court.
The Court made no criticism of the age limit in the Act but Mr Swinney said Labour’s suggestion would be considered as part of the “extensive consultation” he will undertake to improve named person.
Labour education spokesman Iain Gray told BBC Radio Scotland’s Good Morning Scotland: “Some of the training materials produced by (Scottish Government) officials have helped those who want to present the policy as being intrusive and absurd.
“The proportionality of it has been lost and that is why we are saying the government has to take time to win those arguments again, and 16-18-year-olds is very much part of that.
“A 16-year-old should be treated as an adult in so many ways, they can leave school, they can work, they can marry.
“To change this now would be a strong signal from John Swinney that finally the government are listening to critical friends of the policy, such as Scottish Labour, and are willing to go further than simply tweaking the legislation around data-sharing.”
Writing in The Times, Liberal Democrat leader Willie Rennie said: “The government needs to order an independent review to consider the threshold for intervention, the power and duty of the named person, resources, training and data-sharing arrangements.”
Mr Swinney told the BBC: “I’ll certainly give consideration to the suggestion that has been made by Iain Gray and the Labour Party.
“In the process of addressing the implications of the Supreme Court judgement, I want to make sure that we address the concerns and the issues that were raised about the guidance that Iain Gray was referring to, and obviously I will give consideration to this particular point.
“I have indicated that I would refresh the guidance that was available for the named-person policy, because I accepted as a consequence of some of the political and media traffic around this issue that we needed to reaffirm to members of the public the importance of the named-person policy.”
He said it was “a fair assessment” to say there has been a lack of clear communication about the aims of the policy, and he accepts the responsibility for that.
He stressed compliance with the named-person scheme is voluntary but acknowledged the Supreme Court’s recommendation that the opt-out procedure should be made more explicitly clear.
Named person was due to be rolled out at the end of August but Mr Swinney said it has now been delayed indefinitely.