Labour has told the home secretary, Theresa May, that she will have to make further concessions over her snooper’s charter if the party is to support the bill in key Commons votes on Tuesday.

The declaration from Keir Starmer, Labour’s spokesman on the investigatory powers bill, came as the SNP announced it would vote against the measure, saying some of its powers were “unacceptably intrusive and practically unworkable”.

May has already made concessions including tabling a new privacy clause and ordering an independent review of bulk surveillance powers in the bill. She has indicated she is likely to consider backing a Labour amendment ensuring the powers cannot be used to monitor legitimate trade union activity.

Starmer, a shadow Home Office minister, said: “The home secretary has already made significant concessions to meet the improvements called for by Labour. But she needs to go further and concede in the other remaining areas of concern.”

Labour wants an “overarching privacy presumption”, stronger thresholds for accessing internet connection records, stronger protections for journalists and lawyers, clearer restrictions on access to medical records and more robust scrutiny of the judicial process to authorise interception warrants.

A decision by Labour to either support or oppose the investigatory powers bill is likely to provoke a small-scale rebellion in its ranks when a report-stage vote takes place on Tuesday. A group of former Home Office ministers have vowed to back the bill regardless, while a decision by the party to support it would be likely toupset Labour’s civil libertarian wing.

A debate at the parliamentary Labour party on Monday night is to be followed by a shadow cabinet decision on Tuesday.

Joanna Cherry, the SNP’s home affairs spokeswoman, said May’s concessions were small comfort given the bill’s continued lack of adequate safeguards and oversight.

“Last week I wrote to the home secretary listing what would need to be done before the SNP could contemplate giving this bill our support. I have yet to receive a reply,” she said.

“We will be seeking to remove internet connection records and bulk powers from the bill. We have tabled amendments to improve the safeguards afforded by judicial authorisation of warrants, to afford extra protection to citizens who communicate with journalists, lawyers or parliamentarians, to tighten up the overly broad hacking powers and to improve oversight and transparency. It remains to be seen what sort of a response our amendments will receive.”

A Home Office spokeswoman said: “We have always been clear that we will listen to the constructive views of politicians from all sides of the house to ensure the passage of this important bill. We have said the government will be bringing forward amendments at report stage and are willing to consider amendments that are in the interest of both improving the bill and of demonstrating the necessity of the powers it contains.”