The Centre for Legal Aid, Assistance and Settlement (CLAAS), which is funded by UK charity Jubilee Action, received two threatening telephone calls from individuals claiming to represent Al-Qaida on 25 and 26 September.
They came in the same week as the murder of seven workers at aid agency the Christian Peace and Justice Institute in Karachi.
CLAAS has temporarily closed its offices for security reasons.
Jubilee Action works with the centre to provide support for the families of Christians imprisoned or facing execution under Pakistan's punitive blasphemy laws. One Christian on blasphemy charges, Ayub Masih, was released earlier this year after a campaign by the centre and Jubilee Action's sister charity, Jubilee Campaign.
Joseph Francis, the centre's co-ordinator, held a one-day hunger strike in protest at terrorism against Christians in Pakistan.
Ann Buwalda, Jubilee Campaign's US director, said: "These threats have made the difficult but vital work that Joseph Francis and his team are doing even more dangerous. We are looking to the Pakistani authorities to send a message to the extremists that they cannot act with impunity."
A member of UK charity the Church Mission Society was injured in an attack on a missionary school in Murree, Pakistan in August in which six people were killed, but a spokesman for the charity said it did not issue directives on whether workers should leave dangerous countries.
"A number of people have come home but many are staying on,
"Solidarity is important. People take a risk in staying and we salute them for doing so."
Save the Children said it was reviewing security at its office in Karachi in the light of the attack on the Peace and Justice Institute.
"We feel there is an increased level of threat. The Peace and Justice Institute was not an evangelical organisation, just an aid agency,
said a spokesman. Measures will be taken to make the site a more difficult target, including removing identification markings from both vehicles and the office itself.
VSO decided to withdraw from Pakistan in May, although it hopes to return when threats to security have subsided.