A radar scan of the sarcophagus has already indicated the presence inside of three "box like" shapes. The searchers believe these could contain documents and a further examination is now being proposed which they hope will finally prove the link between Greville and Shakespeare.
The initial search, using ground penetrating radar, was approved by the parochial church council and the diocesan council. The team now wants to use an endoscope – a tiny video camera on a long thin tube – to be inserted into the monument to test his claims.
The work would be supervised by Professor Warwick Rodwell, consultant archaeologist to Westminster Abbey, who is keen for the project to go ahead.
Experts said that any manuscripts inside might have disintegrated over the years but could have survived if they are, for example, in "lead-lined boxes", which were common at the time.
The parochial council also wants the sarcophagus to be opened because it believes that any new evidence will bring extra visitors and save the church, the foundations of which date back 900 years, from bankruptcy.
"St Mary's is a beautiful church but is in desperate financial straits," a spokesman said. "Any manuscripts that are found would safeguard its future."
However, the diocesan advisory committee and the Church Buildings Council are resisting the new search, on "ethical grounds" and a final decision could now be taken by the diocese's consistory court.
The search has been prompted by the work of the historian AWL Saunders. He believes there are several clues suggesting Greville, who is a distant ancestor, is responsible for writing a number of Shakespeare's works.
Greville was an eminent dramatist and poet himself, as well as a favourite of Queen Elizabeth I and Chancellor of the Exchequer under James I.
Analysis of the biographical details of his life and the style of his known writings show a very close match to those of Shakespeare, suggesting they could be the same person.
They lived in the same street, had the same friends – including Christopher Marlowe and Francis Bacon – and enemies and were member of the same literary circles. Greville also said in his writing that he was "the Master of Shakespeare".
Mr Saunders has also analysed Greville's work, The Life of Sidney, about his friend Sir Philip Sidney, another prominent courtier, and found what he claims to be clear hints that he wrote Antony and Cleopatra and that he had it and other works, including a biography of Queen Elizabeth, placed in the sarcophagus.
In his book, Greville wrote that he had written the play and had given it "a much more honourable sepulture, than it could ever have deserved" – which Mr Saunders believes is a reference to his ornate sarcophagus. Greville himself was buried in the crypt below the monument.
Greville also wrote to those asking why he had not written a story of Queen Elizabeth's life: "Let him receive this answer from a dead man, because I am confident that no flesh breathing – by seeing what is done – shall have occasion to ask that question whilst I am living."
This is another hint from beyond the grave, according to Mr Saunders, that Greville had buried important documents.
Mr Saunders said: "Fulke spent the equivalent of £300,000 today on a marble sarcophagus at St Mary's. No man would build something like that and leave it empty. There is definitely something down there and we want to find out what it is."
In another intriguing link, Mr Saunders said that Greville was involved with the Rosicrucian (Rose Cross) Order, a secret society of mystics which existed in England in the 16th and 17th centuries.
One of three swords originally on the tomb in St Mary's church and now in Warwick castle appears to have the Rosicrucian symbol.
Many eminent Shakespearean scholars agree that the author of The Tempesthas a deep knowledge of Rosicrucian philosophy.
In 1990 a study by the University of California and Los Angeles compared the scripts of Shakespeare to contemporary Elizabethan writers and Greville was the only one it did rule out as the author of the bard's plays.
The authorship of Shakespeare's oeuvre has been debated since the mid-19th century.
Many argue that the works attributed to him display an understanding of law, history and mathematics which a mere commoner from an illiterate household in Warwickshire could not have had.
The bard's detailed will, in which he notably left his wife "my second best bed with the furniture", fails to refer to any theatrical legacy.
Debates about authorship of the works are recorded as early as the 18th century but began in earnest in the mid-19th century.
Candidates put up by those convinced that William Shakespeare was just a pen-name include Christopher Marlowe, Francis Bacon and the Elizabethan nobleman Edward de Vere.
Shakespeare experts were intrigued by Mr Saunders' claims. Professor Kate McLuskie, the director of the Shakespeare Institute, said: "If they did find a manuscript, that would be wonderful since we have no manuscript of any of Shakespeare's plays. It would keep the Shakespeare industry going for years."
Dr William Leahy, who runs the MA programme in Shakespeare authorship studies at Brunel University, said: "We have had tombs dug up in the past and nothing was found, but we can't make a judgement until the tests have been carried out."
