The threat was made on Monday, where a man called Ms Allan's Telford office leaving a voicemail.
A member of her staff then reported the voicemail, which was left anonymously.
On the voicemail Ms Allan said the caller swore at her and called her an offensive name before claiming someone was going to kill her.
She said the message concluded with the man saying "hope you die".
Helen Blake, spokeswoman for West Mercia Police, confirmed the force is investigating the call.
She said: "We received a report of threats to kill on Monday, December 7.
"The report relates to comments left on a voicemail message and is currently being investigated."
This comes after the Conservative MP was accused of doctoring a constituents email on her Facebook page.
She then became the centre of a social media row after it was claimed that she had added the words "unless you die" to the end of a message she had posted.
The controversy surrounds a message on the MP's Facebook page, attributed to "Rusty, from Dawley", in which he described her as a "robot" with "no empathy for anyone but your super rich buddies and benefactors".
The post concluded by stating "unless you die".
Ms Allan said that she had published the message to illustrate the abuse that MPs are subjected to.
However, the man responsible for the initial email, writing under the pseudonym Rusty Shackleford, immediately replied to the social media post stating that he had not made the final comment.
Ms Allan has now made full statement about the situation.
It says: "There has been confusion and misreporting surrounding recent social media exchanges involving me.
"Some headlines and stories have claimed that I invented fake death threats and that I admitted to doing so.
"This is emphatically not the case and not something which I have ever done, or would ever do.
"The facts are that West Mercia Police are currently investigating a death threat made by telephone to my Telford office.
"The anonymous male voice message was reported to police by a member of my staff.
"At the same time, the Metropolitan Police are investigating a cyber harassment campaign, which has made it difficult for my office staff and me to carry out our roles and duties in Parliament and in Telford."
She continued: "Equally, I have acknowledged that a recent Facebook posting made by me drew upon two anonymous communications sent to me. In conflating elements of the two communications, I created a misleading impression, for which I have apologised. In the climate of intimidation created by the sort of cyber abuse to which MPs of all parties are now commonly subjected, on a daily basis, I believe that this error was understandable.
"However, it emphatically does not amount to my having invented fake death threats or of ever having admitted to doing so. To claim that I did is wholly false. In reporting what I can only believe was a genuine death threat plus an unacceptable level of social media harassment to the police, I believe I have acted responsibly and within my duties as an MP and my rights as a citizen."
The Telford office was closed on Tuesday but has since reopened, however, Ms Allan's surgery due to be held in Southwater tomorrow has been cancelled.
Ms Allan's Twitter and Facebook accounts were also taken down, but the Facebook page has since been reinstated.