Leftist Libertarians will not welcome an alliance with Conservatives, but the Leftist Libertarians are mostly modern liberals floating around the Left/Center alliance uncertain if they should stand for liberty completely or join modern liberals in sacrificing economic liberty for some vague idea of social justice.
Libertarians in general, however, who are mainly classical liberal types, will see the need for an alliance with fiscal, limited government Conservatives. Any social issues which divide the two political groups can be argued over beers, but there's no need to seek political power to push social issues that have to do with individual free choice regarding a person's moral path. One of the big hang ups in the libertarian/conservative alliance is the libertarian promotion to end the War on Drugs. Maybe some young libertarians are focused on legalization of pot because they like to smoke pot, but to dismiss libertarians as potheads because they promote an end to the War on Drugs is a mistake.
There's more to opposition to the War on Drugs than wanting to roll a joint without worrying about the authority. The issue is one of liberty and limited government, but there are also practical concerns, like the number of people incarcerated who aren't violent, bad people, just because they got caught with drugs that are no worse for a person than alcohol. There's also all the violence and death surrounding gang wars associated with illegal drugs. I don't want to make this about legalizing pot, but I want to get past some of the superficial disagreements between Conservatives and Libertarians. Just because a person promotes limits on government power, such as saying government prohibition of marijauna is as bad as prohibition of alcohol in the 20th century, doesn't present a barrier to a political alliance based on all that the groups have in common.
Rand Paul is trying to bridge the gap, and he's probably more conservative than libertarian. Paul and Justin Amash are two representatives who are leading the alliance, and, along with Mike Lee and a few others, they make a make strong core from which to grow. There's a battle for the political philosophy of the Republican Party between moderates and conservatives, and Centrists outnumber limited government conservatives within the party establishment. Centrists appear determined to push the Party to the Left, which to me means toward statism, central planning, social engineering, etc. If the Republican Party can't become a true opposition party that rejects statism and embraces limited government and economic freedom, then I have no use for the Republican Party. No libertarian should have any use for a GOP that simply wants to compete with Democrats from control of a statist system. If Republicans aren't dedicated to restructuring the statist system, then the're useless, and Democrats may as well keep control until a new party can arise.
I don't even know if the few Conservative/Libertarian types like Paul and Amash can withstand the pressure in DC to capitulate. It's easy to take the pragmatic route and justify selling out because you have to play the game to get in power. We'll see what happens, but right now I'm not confident. Libertarians are far too timid when it comes to activism and fighting for limited government, Conservatives are too eage to find the Center path because they don't see any other pathway to power. If Conservatives and Libertarians formed a real, active alliance, who knows what might happen -- it would have to be better than what has happened in the last few decades. They'll have to articulate a new vision that transcends the welfare state. That's a tall order.