Mr Streeting’s stance has faced some push back from others around the Labour administration who feel it breaches the Government’s neutrality on the issue.
MPs have been promised a free vote on the the Terminally Ill Adults (End of Life) Bill and the Prime Minister has said it is “very important” pressure is not put on MPs to vote a certain way.
Mr McArthur’s Bill is currently going through scrutiny ahead of a stage one vote in the Scottish Parliament.
On Tuesday, campaigners from Dignity in Dying Scotland gathered outside the Scottish parliament to support the legislation, attaching handwritten “dying wishes” to a number of olive trees they placed outside the building.
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As the issue is devolved, Mr McArthur’s Bill will not be legally affected by the process at Westminster but the Orkney MSP discussed Mr Streeting’s interventions and whether they may sway people away from supporting his Bill.
The Lib Dem MSP told the PA news agency: “I think the indication from the Prime Minister is that Cabinet ministers would be allowing colleagues an opportunity to make up their own minds on this and not be kind of lobbying behind the scenes.
“It’s difficult to necessarily square that with some of what the Health Secretary has been saying.
“But nevertheless, that process will run its own course and I’m more concentrated on ensuring the scrutiny process here in the Scottish Parliament on my Bill is as robust as it possibly can be.”
Mr McArthur said he agreed with some of Mr Streeting’s points around the need to improve palliative care.
He said his MSP colleagues tend to be “looking for reasons to support rather than excuses to oppose” his Bill.
The UK Health Secretary has asked his officials to carry out a cost analysis of any change and has suggested it could cost the NHS more if a new law was brought in.
Ally Thomson, director of Dignity in Dying Scotland, said public opinion supports a change in the law.
She said: “I think Wes Streeting is coming from a personal point of view.
“I’m very sad we weren’t able to convince him that this isn’t the best thing for people.
“Actually, instead of the fear scenarios that people like Wes may feel, what we see when assisted dying has been introduced in other countries is an investment in palliative care.”
Miro Griffiths, spokesman for the Better Way campaign group which opposes assisted dying, said: “In a society that’s riven by inequality, assisted suicide would be a dangerous step.
“We believe there’s a better path forwards for Scotland.
“This involves ensuring that those living with terminal illnesses have the support they need.”
