Mr Swinney was appointed to the role yesterday with a specific remit to take on the implementation of the controversial plans.
Mid Scotland and Fife MSP Liz Smith is now calling on him to answer the concerns raised by teachers, parents, council leaders, police and social workers over the practical implementation of the scheme.
The Scottish Conservatives have backed a full repeal of the scheme before it is implemented in August.
During the election campaign, other political parties have also raised concerns over how it might operate, and have backed a pause.
The Scottish Conservatives say today that - leaving aside the principled arguments around the scheme - the SNP must now engage with genuinely held concerns over how it is going to work.
Scottish Conservative MSP Liz Smith said:
“We welcome the fact that John Swinney has been put in charge of dealing with named persons.
“As everyone knows, we are opposed to the named person legislation in principle and we will continue to support a full repeal.
"At the same time, however, we also believe the SNP must answer the concerns of professionals in the field over its impact on their workload and how it is going to operate.
“Quite simply, we believe the scheme is unworkable, and will have a massively detrimental impact.
"It is going to pile on mountains of extra work on the shoulders of teachers, health visitors and care professionals. As council leaders have warned, it may even put people off applying for senior teaching posts.
“Up until now, the approach from the SNP has been to brush these concerns aside as being misplaced, and accuse critics of scaremongering.
"I hope John Swinney will press the reset button and accept that people’s concerns are genuine and based in fact.
“With the election campaign now over, the SNP needs to reflect and reconsider. It’s time for the SNP to engage.”