Glasgow MSP Annie Wells has written to the Prime Minister - the letter she says Nicola Sturgeon "should have written" rather than the one she sent asking for a Section 30 Order for a second independence referendum - asking him to rearrange a promised expert meeting as soon as possible.
The UK government announced in October it would host a summit in Glasgow before Christmas to discuss the issue, but it was postponed due to December's snap general election.
Ms Wells called on both the UK and Scottish governments to place the issue at the top of their agendas and to put their political differences aside. She said: "I lost a neighbour. Across Scotland we lost 1,187 people in 2018, and I heard from so many families who lost loved ones in 2019.
"So I've asked the Prime Minister to make the drug deaths crisis his top priority in Scotland. This year we should be focused on saving lives instead of getting caught up in politics and the usual constitutional blame game."
However in her letter she adds: "I know that the UK government is limited in the direct action it can take. The NHS in Scotland in entirely under the Scottish Government's control and they are solely responsible for providing funding increases and improving treatment and rehabilitation services.
"I am hopeful that this proposed summit could see both of Scotland's governments find a way to work more closely together and finally treat the drug deaths crisis with the urgency it deserves."
Scotland's drugs deaths have been described as a "health emergency", with death rates the worst in Europe. Dundee has recorded the highest rate of drug-related deaths per 1,000 population of all council areas in Scotland.
The Scottish Government said it planned to hold a summit on drug deaths at the start of 2020, and that it had "repeatedly" invited the UK government to attend but that, to date, they had refused.
Nicola Sturgeon raised the issue of drugs deaths with Mr Johnson at their first meeting last year, and the Scottish Government has been urging ministers at Westminster to change the approach on drugs misuse from a judicial matter to one of public health.
SNP MPs have also pledged to issue fresh demands for powers over drugs laws to be devolved to Holyrood "if the UK government continues to refuse to act".
Alison Thewliss MP said: “Scotland faces a public health emergency and now even a Tory MSP is calling on the UK Government to act.
"It’s clear that the current UK drugs laws are not working. We desperately need a fresh approach to tackle this issue head on, but Westminster politicians have so far stood in the way of progress.
“The Tories have consistently displayed a shocking lack of empathy towards this emergency. We don’t need any more platitudes, we need action or we risk an even greater number of drug related deaths next year. Action to allow a medically-supervised overdose prevention room in Glasgow must be an early priority of this Tory government.
“The communities I represent have suffered the impact of problem drug use for far too long - the time for action is now. We need an urgent shift in the law to allow fresh and radical approaches to prevent deaths, and we need that work to begin immediately."
A Scottish Government spokesperson said: "We firmly believe the outdated Misuse of Drugs Act 1971 should be amended to allow us to implement a range of public health focused responses. We have called on the UK government to amend the act or to devolve those powers to Scotland, and this must be part of any discussion we have."
A spokeswoman for the Home Office said the number of drug deaths across the UK was "extremely concerning", in particular the figures for Scotland. She said improving access to treatments such as Naloxone - used to treat overdoses of methadone, morphine and fentanyl - was key.
She added: "We will continue to work with the Scottish government to tackle drug-misuse and harm and sustain our support for programmes which reduce the health-related harms of drugs, such widening the availability of Naloxone to prevent overdose deaths."