Any suggestions on how to deal with client who has a loyalty to your competition?

They are probably loyal for a good reason. Can you give them an even better reason to break that loyalty, with all the inherent risks associated in changing suppliers?

A good place to start would be to understand the sources of that loyalty. How did the relationship evolve? What do they most value about the competitor? And if there was one thing that competitor could do to improve, what would it be?

It's not clear what you are trying to sell to them, but if it's any halfway complex product or service, I'd start by sharing information that you hope they might find valuable before you try and sell to them. Build the trust first. If you can't, what on earth makes you think you can displace the incumbent?

p.s. I don't think the answer can be found in a "killer script". In fact, I am absolutely convinced that you can't.

First of of all dealing with a customer who is loyal to a competitor requires a lot of patience from the salesperson. The salesperson has to act like someone besieging a fortress. The fortress (i.e. the loyal relation between customer and competitor) has to be carefully observed for finding small cracks that can be widened and then be used as targets for an attack. A besieger can hardly mobilize enough force for a frontal attack with brute force to be successful .
Examples of cracks in the fortress could be less than optimal customer experience, obsolescence of competitive products, total cost of ownership for the competitor's solution do no longer match industry benchmarks, the competitive solution is written off in the books of the customer or a lease comes to an end. .

When considering an attack on price, it must further be considered that the there are switching cost for a customer (e.g. re-training users, refurbishing site, residual amortization integartion cost with adjacent systems etc. ). Small differences in purchasing prices are thus not a motivator for a customer to break loyalty with the incumbent supplier.

What I would certainly not do is trying to bad mouth the competitor. This is like saying to the customer that he/she made a bad original choice. I doubt this is a good start for a relationship trying to shift loyalty towards you as it is hardly a recipe to build trust.

I share Bob's view that the answer to this question cannot be found in a "killer script".

Overcoming “Loyalty” to a Competitor is difficult and complex. I agree with Christian “knocking” the Competition, however subtly, will be counter-productive.

Customer Loyalty is essentially an emotional aspect; it is seldom listed in their Rational Decision Criteria. Yet, it can be hidden behind the “Do Nothing” option where it appears as “affiliation with current supplier”.

Often, Loyalty is rooted in a “Good Relationship” with the Competitive Salesperson, or the multiple Contact points in the incumbent Supplier. “I know who to call!”

Having sold head to head against IBM for 20 years, we became very familiar with the “Loyalty” Objection, which could be expressed as “We only buy from IBM” or “Nobody has ever been fired for buying IBM”. Either way we were not going to get the deal.

Dealing with the Loyalty Objection. No scripts that I know of but the process is first turn the objection into a Question.
“If IBM don’t make the best fit solution would you then consider other providers?”

Then, secondly minimise the objection: “By showing independence from IBM and using another supplier ‘on this occasion’ would that not cause IBM to redouble their efforts to win you back?

Thirdly, ‘demonstrate’ best fit to their situation, usually both functionally and financially.
To do this you need to know:
How to make “Benefit” Statements
(as Defined by Rackham in SPIN&FAB) and
How to make “Value Propositions”
(as Defined by McKinsey).

This is "Why they should overcome their natural loyalty and instead exercise choice" to come to you.
If you are not best fit to their Actual Decision Criteria it won’t work!
If you don’t know the difference between a feature and a benefit it won’t work, if you don’t know the difference between a financial ‘claim’ and a Value Proposition, it won’t work.

Finally, and this is where most Sales fail, you need to offer PROOF for all that you have said. Show Testimonials from other Customers who changed supplier, and saw an improvement in relations with their old provider. Proof through Demonstration or Reference that your Benefits and Values do come from your Products or Services. And finally you may have to warrant your claims, the money back guarantee!

If after doing all of these well and yet it still doesn’t work, then you have a weak Relationship with the Prospect. Lose the sale gracefully, ask for a lost business review with the Prospect and then go back to relationship and trust building for your next opportunity.
Good Luck!