The Deputy First Minister was speaking on the BBC's Good Morning Scotland radio programme on Friday when said he hopes Scotland and the UK can remain in lockstep over lockdown measures.
Mr Swinney branded messages coming from Westminster to English media as "disastrous" and a "foolish mistake to be made".
His comments comes the day after Foreign Secretary Dominic Raab spoke at the daily government briefing, where he said the UK can start to think about the next phase in the pandemic.
He said: "The Prime Minister's words were in stark contrast to the front pages of most of the newspapers down south yesterday which had clearly been briefed by somebody within the UK government that there was going to be a significant relaxation of the lockdown - and that was entirely the wrong message and a foolish mistake to be made."
Mr Raab had said at yesterday’s Downing Street press conference: “The virus is not beaten yet.
“It remains deadly and infectious, and we are working very hard right across government and with local government to bring it down in areas of concern, like in care homes, and I’m confident we can do it and we will do it.
“But, because we held firm three weeks ago, we are now in a position to start to think about the next phase in this pandemic.”
Asked about the apparent delays for some key workers to be tested for Covid-19, Mr Swinney told Good Morning Scotland: “One of the issues that I would have to say has frustrated us at times, is that we have had more testing capacity available than has been used.”
He added: “I can assure you that the issues that you're raising - the practical, real, legitimate issues - need to be addressed, so that we can have an efficient testing strategy.
"For somebody to be told: "Look, there's a test available at Arbroath" and you're living in Glasgow, is just not acceptable. I totally recognise that.
"So we have to make sure that the capacity is used to the full throughout the country.
“I live in the Tayside area and I've seen NHS Tayside really stretching the delivery of tests and reaching out to organisations asking them, 'do you have anybody that needs a test? Get them in here and we can get them tested'.
“There are really proactive steps being taken in different parts of the country to reach different groups and to make sure they can be tested, and perhaps we need to have a bit more of that in the country.
“But we'll certainly reflect on those points in the course of the review undertaken today.”
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