Speculation that a late challenger might still emerge in the increasingly bitter race for the Republican presidential nomination is set to surge after former Florida governor Jeb Bush made remarks criticising the current field.

Bush, who is the brother of President George W Bush and son of President George Bush Sr, is a beloved figure among many conservatives who see him as a strong and charismatic leader who is popular in the must-win swing state of Florida.

That contrasts with a widespread unease among many Republican leaders and grassroots activists with the remaining crop of Republican candidates and the vitriolic nature of the fight between frontrunner Mitt Romney and his main challengers Rick Santorum and Newt Gingrich.

In answers to questions from the audience after a speech in Dallas on Thursday, Bush cautioned the remaining Republican campaigns from drifting so far to the right that they put off the key independent voters needed to beat President Barack Obama in November.

"I think it's important for the candidates to recognise though they have to appeal to primary voters, and not turn off independent voters that will be part of a winning coalition," Bush told the audience according to CBS news.

Bush also directly took on the strident tone of recent Republican debates, accusing participants of scare-mongering. "I used to be a conservative and I watch these debates and I'm wondering, I don't think I've changed, but it's a little troubling sometimes when people are appealing to people's fears and emotion rather than trying to get them to look over the horizon for a broader perspective and that's kind of where we are," he said according to Fox News.

With Mitt Romney failing so far to secure the nomination but with no convincing challenger emerging to unseat him, many Republican pundits have speculated about the possibility that none of the current field will be able to amass enough support to secure the nomination this August in Tampa.

Though that is still unlikely, and Romney remains favourite to win the contest, it has led to a slew of names being mentioned as possible "white knights" who could still enter the race or emerge at Tampa as a compromise candidate to unite a splintered party. They include Bush, New Jersey governor Chris Christie, Indiana governor Mitch Daniels and Wisconsin congressman Paul Ryan.

Though none of these figures have expressed any intention to run, and several have repeatedly denied it, Bush's comments are likely to set the rumour mill spinning furiously.

They also come after Tea Party favourite Sarah Palin entered the fray, raising the idea that she might see herself as her party's saviour. In recent interviews the former Alaska governor has said she would "help" out the party if a contested convention happened and told CNN earlier this month that she believed such an event would be a good thing. "I don't think it would be a negative for the party … That's part of the competition, that's part of the process and it may happen," she said.

Ron Paul's campaign has also complicated matters. Though the libertarian-leaning Texan congressman has not yet won a single state's popular ballot, he is trying to build up a large number of delegates to take to Tampa. In caucus states, where complex rules mean the number of delegates assigned to a candidate can outweigh their score in the popular vote, Ron Paul's campaign is working hard to win as much support as possible. That could see him amass a body of delegates in Tampa that far exceeds his standings in the polls and makes a contested convention, with no one having enough support to secure victory, more likely.