The provider has increased the annual interest rate on its monthly saver account from 2% to 3% to new customers, who must hold a current account with Lloyds Bank in order to get the deal.
There have been signs of banks and building societies increasing competition after a new industry-wide customer guarantee was introduced earlier this month to make it easier for current account holders to ditch their old bank and switch to a new one.
Under the new rules, the length of time it takes to switch current accounts has been cut from up to 30 working days to just seven, and all existing outgoing and incoming payments automatically follow the consumer to their new provider. Consumers are also entitled to refunds for money lost as a result of anything going wrong with the switching process.
Rachel Springall, spokeswoman for financial information website Moneyfacts, predicted that more banks will start to follow in Lloyds' footsteps by improving products which are tied to their current accounts to produce an overall "package" of deals which will encourage consumers to switch.
Several providers are already offering incentives for people to switch in the form of cash or vouchers. First Direct is currently offering new customers £125 while Halifax is offering £100. M&S Bank is offering people £100-worth of Marks & Spencer gift cards when they switch.
Lloyds Bank relaunched on to the high street as a standalone brand for the first time in 18 years yesterday. The "revitalisation" of Lloyds means the bank, which has a 250-year history and 1,300 branches across England and Wales, has changed its branding to reflect the dropping of TSB from its name.
The new TSB bank appeared on the high street earlier this month after Lloyds offloaded 631 branches and eight million accounts to meet European competition rules.
The new gross rate of 3% AER (annual equivalent rate) on Lloyds' monthly saver account is fixed for 12 months from the account being opened. Interest is paid upon maturity and after the initial 12-month period the rate reverts to 0.75%.
Customers holding the savings product must pay a deposit of at least £25 a month by standing order and the maximum deposit is £250. Withdrawals are allowed.
The new 3% rate comes at a time when savers are still struggling to find accounts with any decent returns and the bank of England base rate is expected to remain at a record 0.5% low for some time to come.
Ms Springall said: " It's great to see competition in the regular savings market, these accounts are ideal for people looking to save little and often who can commit to a set savings plan.
"An interest rate of 3% is significantly higher than many other savings vehicles on the market, such as easy-access, notice and short-term fixed bonds.
"However, customers would be wise to check the latest best deals available because First Direct, HSBC and M&S Bank customers can get 6% on a fixed regular saver if they hold a current account."
She added: "Loyalty doesn't always pay, so it's worth checking out what's on offer elsewhere. This is especially important after a deal ends as savers are usually moved on a low-interest easy-access account."