The High Court in Glasgow was told that Miss Spence was trying to make an appointment with the Queen's bank in relation to bearer bonds.
Detective Constable Alistair Brown, 50, one of the team of officers from Strathclyde Police investigating Miss Spence's disappearance, said that he phoned the London bank after seeing the e-mails.
DC Brown was giving evidence at the trial of Colin Coats, 42, David Parker, 38, Paul Smith, 47, and Philip Wade, 42 who all deny abducting, torturing and murdering Lynda Spence in a flat at Meadowfoot Road, West Kilbride, between April 14 and 28, 2011.
He told Solicitor General Lesley Thomson QC, prosecuting, that Miss Spence sent an e-mail to the bank on April 11, 2011 claiming she had inherited bearer shares and received a reply the next day.
In her e-mail Miss Spence says: "Hi, John hope you are well. Wondered if you had managed a meeting for me."
The reply from the bank states: "Please let me know who you bank with. I will arrange for Coutts' personal banker to call you back. As regards to bearer shares you have recently inherited, I would need to know who they are issued to and which currency and the value."
The jury has already heard that Miss Spence was involved in a scheme involving fake Danish Bearer Bonds which she claimed were worth £2.8million.
DC Brown was then asked if he was involved in any searches and replied that he had been involved in the search of premises in Glengarnock in 2012.
He was asked by Miss Thomson: "Was that the only furnace that was searched," and replied: "As far as I know, yes."
The jury has heard from call centre manager Peter Haddley, 26, who knew Coats in prison and claimed that he had said he killed her, cut off her head and put her body in a furnace.
The trial continues.