* Law promises savings on carbon, energy, cheaper fuel
* Cost to energy companies estimated to be very low
* Objectors say new rules could curb growth, cost too much
BRUSSELS, June 4 (Reuters) - EU president Denmark is pressing European lawmakers and senior politicians to compromise over an ambitious energy saving law designed to help meet a 2020 EU target to cut consumption by a fifth, but which critics say would stifle growth.
Denmark made energy efficiency a priority for its six-month tenure at the head of the European Union and wants to nail down an agreement before handing on to Cyprus at the end of June.
The law would enforce energy saving through measures such as improved technologies and better insulation for public and domestic buildings.
Supporters say it could lower energy bills and cut fuel imports, which last year hit 315 billion euros ($389.43 billion) for oil alone, according to the European Commission. But opponents say the measures require investment upfront and are unreasonably prescriptive.
Negotiations to agree a text, bringing together the European Union's three decision-making bodies - the Commission, the Parliament and the Presidency, representing the member states - resume on Tuesday, with a final round expected on June 13.
"We still believe an agreement is possible that would make a significant contribution," a source close to the Danish presidency said.
"But we need everybody including the European parliament to show flexibility at this stage."
Many in Brussels believe the 27-nation bloc will only achieve cuts in energy use of around 10 percent by the end of this decade. Supporters of the Energy Efficiency Directive (EED) pushed by Denmark say it will help get the European Union back on track to achieve the 20 cut target, provided a sufficiently ambitious text can be agreed.
However, member states are reluctant to invest in the current climate of austerity, even if it would result in long-term savings.
Britain opposes what it sees as unnecessary regulation and, together with nations such as Austria and Germany, argues that savings already made should taken into account as well as any new ones, EU sources say.
According to the Commission, improved efficiency could create around half a million jobs and boost the economy by 34 billion euros ($45 billion) in Gross Domestic Product by 2020.
It has also put the cost to the energy companies at only one euro cent for every kilowatt hour of energy saved.
Research released on Monday suggested the Commission's figures were too timid on the benefits of improving the bloc's 2020 energy saving - from the 10 percent expected under business-as-usual to 20 percent under the proposed law.
"Put simply, for every one euro of energy cost saving, an additional one euro could be saved due to lower energy prices," Dutch consultancy Ecofys said.
"Therefore net annual cost savings of the order of 100 billion euros can be expected on top of the 107 billion that will result from implementing cost-effective energy savings measures."
In addition, Ecofys predicted consumers would benefit from cheaper energy, as lower demand and less need to invest in major infrastructure upgrades should help to drive down prices.
($1 = 0.8089 euros) (Editing by Jon Boyle)