Fenton has been charged with possession of banned substances including anabolic steroids, with the case adjourned until after the festival on 20 March.
His entries include Last Instalment, third favourite for the Gold Cup.
"We want to take steps which serve to uphold public confidence in the races," said BHA chief executive Paul Bittar.
"Any development or set of circumstances which brings the integrity of our sport into question is of considerable concern."
The BHA is putting greater resources into anti-doping measures after completing a review of its control procedures in December.Fast-track testing of the blood and hair samples from Fenton's horses will take place at HFL Sport Science, Newmarket, with the results of the tests available next week.
That followed the handing out of bans to two high-profile trainers for doping offences during 2013 - Mahmood Al-Zarooni was banned for eight years and Gerard Butlerfor five years.
"The events of 2013 highlighted the need to increase the deterrent against the misuse of drugs and medication, and in particular anabolic steroids," Bittar said.
"Since then significant steps have been taken towards achieving a set of international minimum standards, with a number of jurisdictions adopting a zero-tolerance policy regarding the use of steroids in racing."
In addition to Last Instalment, who recently returned from injury to win the Hennessy, Fenton's horses on track for Cheltenham include former Champion Bumper winner Dunguib.
The Fenton-trained The Tullow Tank, third favourite for Cheltenham's Neptune Novices Hurdle, will miss the race on the orders of owner Barry Connell.
Fenton, who will also be interviewed by the BHA at his base in County Tipperary, Ireland, had his yard inspected by the Irish Department of Agriculture in January 2012.
Bittar added: "Although the inspection at Philip Fenton's yard took place in Ireland over two years ago, and therefore before recent upgrades in deterrents in this country, there is no room for complacency.
"We consider it incumbent upon all involved with British and Irish racing, not just the authorities, to demonstrate their opposition to the misuse of drugs and medication, in particular anabolic steroids, and be open and transparent about all practices."