Reprieve has filed an official complaint with the government over the firm's refusal to "come clean" about its role in the strikes.
Concerns centre on BT's role in providing communications infrastructure between a US base in Britain and Camp Lemonier in Djibouti, from where the US launches its drone attacks on Yemen.
The CIA and US special forces have carried out numerous covert drone strikes in Yemen and Pakistan, killing hundreds of innocent people.
The strikes are considered to be illegal under international law as neither country is an official war zone.
Now financial firms including Standard Life, Blackrock, ING and Lloyds have said they are re-examining their investments in BT.
Lloyds told Reprieve that it had "asked our investment arm to investigate this matter in full."
Reprieve corporate social responsibility advocate Catherine Gilfedder said: "BT's response to date has been to refuse to address how the US government uses the company's systems.
"If the company is playing a key role in the US's illegal drone war then its investors and its customers deserve to know.
"Hundreds of civilians have died as a result of covert drone strikes, the latest step in the misguided War on Terror. BT must come clean on whether or not it is involved in this."
A BT spokesman told the Star that it had spoken to investors contacted by Reprieve and insisted that the company had "no knowledge of US drone strikes and has no involvement in any such activity.
"BT can categorically state that the communication system mentioned by Reprieve is a general purpose fibre-optic system.
"It has not been specifically designed or adapted by BT for military purposes, including drone strikes.
"BT takes its human rights obligations very seriously."