Ms Callaghan said East Dunbartonshire Council has yet to give a clear indication on whether they will take part in the Afghan Relocations and Assistance Policy.
Up to 550 Afghan refugees initially are likely to be resettled in Scottish communities as part of the UK government scheme to protect vulnerable people fleeing the country.
The move is designed to safeguard those identified as most at risk. The UK has committed to taking about 5,000 people in the first year.
Ms Callaghan, along with Kirkintilloch East MP Stuart McDonald, wrote to council leaders Andrew Polson and Vaughan Moody last week urging them “to take swift action and participate in the Afghan Citizens’ Resettlement Scheme, following an outpouring of emails from our constituents.”
The local MP said yesterday (Wednesday): “I thank Councillor Moody for replying so promptly to our letter. I've subsequently been liaising with the chief executive of the council on this matter to press for an urgent resolution.
“Stuart and I have yet to receive a clear indication on whether the council will take part in the Afghan Relocations and Assistance Policy.
“I have stressed the importance of the situation and that no delay or hesitation is acceptable amidst such a tragic humanitarian crisis.
“Scotland - as we always do - stands ready to provide refuge and shelter for those fleeing unimaginable terror. Stuart and I will work alongside the council to ensure those who arrive on our shores know East Dunbartonshire is their welcoming home.”
Councillor Andrew Polson, Joint Leader of the Council, said: "We have received the MPs' letter and share their concerns. We are currently engaged in ongoing national discussions, involving COSLA, to agree a co-ordinated, effective response as soon as possible."
Councillor Vaughan Moody, Joint Leader, added: "We are awaiting information and work will take place as required with a wide range of partners, as it did when families of refugees were welcomed locally in 2017 through the Syrian Vulnerable Persons Resettlement Scheme."