Master of the Rolls Lord Neuberger believes that allowing broadcasters to televise court cases could make the public more engaged in the process.
However, he said that judges should always retain the veto over what aspects are aired and he also stopped short of recommending that criminal trials should be televised.
Currently, only the proceedings and judgements of the UK Supreme Court can be shown on television, despite a sustained campaign to relax the rules.
Last December, John Ryley, the head of Sky News, called for TV cameras to be allowed in courtrooms to help tackle the "growing public dissatisfaction with the judicial process".
Ryley said that banning cameras from court hearings makes the public suspect "that something is rotten behind those closed doors".
However, opponents of the approach have expressed concern that it could turn high-profile trials into a media circus.
In a speech yesterday to the Judicial Studies Board, Lord Neuberger stressed that it was a long-standing legal principle that justice should be carried out in public.
He also suggested that hearings from the Supreme Court and the Court of Appeal should be made available on a judicial version of the BBC's Parliament channel, or via BBC iPlayer.
"If we wish to increase public confidence in the justice system, transparency and engagement, there is undoubtedly something to be said for televising some hearings, provided that there were proper safeguards to ensure that this increased access did not undermine the proper administration of justice," he said.
"Such an idea would have to be looked at very carefully, and it would not be sensible for me to try and make any firm suggestions. But, if broadcasting of court proceedings does go ahead, I think it would be right to make two points, even at this tentative stage.
"First, the judge or judges hearing the case concerned would have to have full rights of veto over what could be broadcast. Secondly, I would be very chary indeed about the notion of witness actions or criminal trials being broadcast - in each case for obvious reasons."
A 2004 government study on televised court hearings indicated that 55% of respondents supported a complete ban on broadcasting criminal cases, but 45% believed that cameras should be allowed to cover parts of a trial.
Broadcasters have previously worked to circumvent the ban on courtroom cameras for major cases, such as Sky News using 3D digital graphics to reconstruct 2003's Soham Murders trial of Ian Huntley and Maxine Carr.
In January, STV covered the sentencing of disgraced Scottish politician Tommy Sheridan by using Twitter as a way to get around the television boycott.
Lord Neuberger endorsed the provisional decision to allow courtroom tweets, arguing that it is a way to allow journalists to "unobtrusively" report on court proceedings.
"Why force a journalist or a member of the public to rush out of court in order to telephone or text the contents of his notes written in court, when he can tweet as unobtrusively as he can write?" he said.
"Whatever the outcome of the consultation, I doubt, however, that we will see the development of tweeting from the bench."