The former England captain, who went on to coach Gloucester and Wasps, is currently president and chief executive of USA Rugby.
The RFU are looking for a new chief executive after John Steele was ousted from the position on Friday, largely as a result of the flawed recruitment process for the new performance director.
In a Twitter message to RFU council member Andrew Sarek, Melville said: "I will apply for the ceo job, its what I have been doing for the last 4 years."
Steele had only been at the helm for nine months but the RFU board felt they had lost confidence in his leadership and were concerned by feedback they received from staff and key stakeholders.
RFU chairman Martyn Thomas has already spoken to the recruitment consultant he wants to head up the search for a new chief executive.
Simon Halliday, a former RFU councillor, and Malcolm Wall, who was beaten to the post by Steele, have also been mentioned as potential candidates for the job.
Melville was on the RFU's early shortlist for the performance director position but withdrew from the running after deciding it was destined for Sir Clive Woodward.
The RFU are yet to deliver on that prediction - but they appear to be getting closer.
Thomas has opened the door for Woodward's return to Twickenham by inviting him to get back in touch over the performance director post.
Woodward walked away for the recruitment process following the RFU's indecision about whether the job would include influence over the England team.
Steele cancelled Woodward's interview on March 17 and then decided to downgrade the position.
Woodward has publicly pledged his commitment to the British Olympic Association through to 2012 - but is reportedly still interested in taking up the role at Twickenham.
When that was put to Thomas, he said: "If that is the case then Clive and I need to speak. Clive knows my telephone number and I would happily speak to Clive Woodward.
"You have got to respect what the man has done. This guy has a unique set of skills.
"He not only understands rugby at the elite end but he has developed a talent for sporting success across a variety of disciplines. You have to see him."
Thomas confirmed the RFU would not consider a job-share arrangement that would allow Woodward to remain involved with the BOA until after the 2012 Olympics, as had been mooted.
"We need a man that has a skill set that is 100% committed to putting England back to where they need to be," said Thomas.
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