"Our focus isn't on any single candidate. Our focus is really [on putting in] place a team that we think is really strong," he said. "I discount nobody."
With fundraising reports due to the Federal Election Commission later this month, Leppert said he's "pushed hard" to have a strong showing in his first quarter in the Senate race.
"That has been very positive," he said, but declined to reveal the figures from his April report.
"We'll release those in the next couple of days," he said. "I've been very, very pleasantly surprised at the reception that we’ve had."
Observers have predicted candidates will need to spend upward of $20 million to be competitive in the primary. Dewhurst and other potential candidates such as Texas Rep. Michael McCaul (R) have personal fortunes to draw on, which could help them gain an advantage.
Leppert, who headed Turner Construction Co. before being elected mayor, said he would contribute some of his own money to his campaign. "I'm going to put my money where my mouth is," he said. "We will be very competitive and very viable as a candidate in fundraising, getting our message out and all the key things that we need to do to eventually win."
Leppert's message will be built around his time in the mayor's office. In his announcement video in February, he talked up the "business principles" and "common-sense conservative judgment" he brought to the job.
"To combat crime, we increased the police force — without raising taxes," he said. "Violent crime: It plummeted 30 percent."
He calls being mayor "probably the most accountable type of job anywhere in the nation." But some observers note that Leppert resigned before his first term ended in order to run for Senate, which could make him vulnerable to the charge that he's a quitter.
Leppert said that won't be an issue.
"When we came in, we laid out some specific objectives and priorities," he said. "I think everybody understands that [in] each one of those areas, we not only completed them, but we exceeded projections.
"I think people feel good about that," he continued. "I think people also understand that although it was a couple months short of the end of the term, the reality was that most of the council races are contested. So from a policy standpoint, there was very little that was going to be accomplished in the next couple months anyway."
Leppert bills himself as a staunch fiscal conservative, and said he would resist voting to raise the federal government's debt ceiling.
"I think we've come to the point where a line has to be stood," he said. "I would not vote for [raising the debt ceiling] unless I knew that there was real discipline that was coming in the system. Discipline in terms of caps, balanced-budget amendments, those sorts of things."
Asked if he would be the deciding vote against raising the debt ceiling, thereby putting the country into default with its creditors, Leppert pointed to other options.
Instead of voting to raise the debt limit, Leppert indicated he would support legislation introduced by Sen. Pat Toomey (R-Pa.) that would allow the government to avoid defaulting on its debts if the $14.3 trillion limit were reached by requiring the Treasury Department to pay principal and interest on public debt before making any other payments. "You don't go into default," said Leppert.
On immigration reform, he said border security should be the priority. "We have to control the borders," Leppert said. "We need to do it in a way that the policy does not provide amnesty."
--Updated at 4:45 p.m.